tl;dr - Many people love 2-node clusters because they seem conceptually simpler and 33% cheaper, but while it’s possible to construct good ones, most will have subtle failure modes
The first step towards creating any HA system is to look for and try to eliminate single points of failure, often abbreviated as
It is impossible to eliminate all risk of downtime and especially when one considers the additional complexity that comes with introducing additional redunancy, concentrating on single (rather than chains of related and therefor decreasingly probable) points of failure is widely accepted as a suitable compromise.
The natural starting point then is to have more than one node. However before the system can move services to the surviving node after a failure, in general, it needs to be sure that they are not still active elsewhere.
So not only are we looking for SPoFs, but we are also looking to balance risks and consequences and the calculus will be different for every deployment 
There is no downside if a failure causes both members of a two node cluster to serve up the same static website. However its a very different story if it results in both sides independently managing a shared job queue or providing uncoordinated write access to a replicated database or shared filesystem.
So in order to prevent a single node failure from corrupting your data or blocking recovery, we rely on something called fencing.
At it s heart, fencing turns a question Can our peer cause data corruption? into an answer no by isolating it both from incoming requests and persistent storage. The most common approach to fencing is to power off failed nodes.
There are two categories of fencing which I will call direct and indirect but could equally be called active and passive. Direct methods involve action on the part of surviving peers, such interacting with an IPMI or iLO device, whereas indirect relies on the failed node to somehow recognise it is in an unhealthy state (or is at least preventing remaining members from recovering) and signal a hardware watchdog to panic the machine.
Quorum helps in both these scenarios.
In the case of direct fencing, we can use it to prevent fencing races when the network fails. By including the concept of quorum, there is enough information in the system (even without connectivity to their peers) for nodes to automatically know whether they should initiate fencing and/or recovery.
Without quorum, both sides of a network split will rightly assume the other is dead and rush to fence the other. In the worst case, both sides succeed leaving the entire cluster offline. The next worse is a death match , a never ending cycle of nodes coming up, not seeing their peers, rebooting them and initiating recovery only to be rebooted when their peer goes through the same logic.
The problem with fencing is that the most commonly used devices become inaccessible due to the same failure events we want to use them to recover from. Most IPMI and iLO cards both loose power with the hosts they control and by default use the same network that is causing the peers to believe the others are offline.
Sadly the intricacies of IPMI and iLo devices is rarely a consideration at the point hardware is being purchased.
Quorum is also crucial for driving indirect fencing and, when done right, can allow survivors to safely assume that missing nodes have entered a safe state after a defined period of time.
In such a setup, the watchdog’s timer is reset every N seconds unless quorum is lost. If the timer (usually some multiple of N) expires, then the machine performs an ungraceful power off (not shutdown).
This is very effective but without quorum to drive it, there is insufficient information from within the cluster to determine the difference between a network outage and the failure of your peer. The reason this matters is that without a way to differentiate between the two cases, you are forced to choose a single behaviour mode for both.
The problem with choosing a single response is that there is no course of action that both maximises availability and prevents corruption.
If you choose to assume the peer is alive but it actually failed, then the cluster has unnecessarily stopped services.
If you choose to assume the peer is dead but it was just a network outage, then the best case scenario is that you have signed up for some manual reconciliation of the resulting datasets.
No matter what heuristics you use, it is trivial to construct a single failure that either leaves both sides running or where the cluster unnecessarily shuts down the surviving peer(s). Taking quorum away really does deprive the cluster of one of the most powerful tools in its arsenal.
Given no other alternative, the best approach is normally to sacrificing availability. Making corrupted data highly available does no-one any good and manually reconciling diverant datasets is no fun either.
Quorum sounds great right?
The only drawback is that in order to have it in a cluster with N members, you need to be able to see
N/2 + 1 of your peers. Which is impossible in a two node cluster after one node has failed.
Which finally brings us to the fundamental issue with two-nodes:
quorum does not make sense in two node clusters, and
without it there is no way to reliably determine a course of action that both maximises availability and prevents corruption
Even in a system of two nodes connected by a crossover cable, there is no way to conclusively differentiate between a network outage and a failure of the other node. Unplugging one end (who’s likelihood is surely proportional to the distance between the nodes) would be enough to invalidate any assumption that link health equals peer node health.
Making Two Nodes Work
Sometimes the client can’t or wont make the additional purchase of a third node and we need to look for alternatives.
Option 1 - Add a Backup Fencing Method
A node’s iLO or IPMI device represents a SPoF because, by definition, if it fails the survivors cannot use it to put the node into a safe state. In a cluster of 3 nodes or more, we can mitigate this a quorum calculation and a hardware watchdog (an indirect fencing mechanism as previously discussed). In a two node case we must instead use network power switches (aka. power distribution units or PDUs).
After a failure, the survivor first attempts to contact the primary (the built-in iLO or IPMI) fencing device. If that succeeds, recovery proceeds as normal. Only if the iLO/IPMI device fails is the PDU invoked and assuming it succeeds, recovery can again continue.
Be sure to place the PDU on a different network to the cluster traffic, otherwise a single network failure will prevent access to both fencing devices and block service recovery.
You might be wondering at this point… doesn’t the PDU represent a single point of failure? To which the answer is “definitely“.
If that risk concerns you, and you would not be alone, connect both peers to two PDUs and tell your cluster software to use both when powering peers on and off. Now the cluster remains active if one PDU dies, and would require a second fencing failure of either the other PDU or an IPMI device in order to block recovery.
Option 2 - Add an Arbitrator
In some scenarios, although a backup fencing method would be technically possible, it is politically challenging. Many companies like to have a degree of separation between the admin and application folks, and security conscious network admins are not always enthusiastic about handing over the usernames and passwords to the PDUs.
In this case, the recommended alternative is to create a neutral third-party that can supplement the quorum calculation.
In the event of a failure, a node needs to be able to see ether its peer or the arbitrator in order to recover services. The arbitrator also includes to act as a tie-breaker if both nodes can see the arbitrator but not each other.
This option needs to be paired with an indirect fencing method, such as a watchdog that is configured to panic the machine if it looses connection to its peer and the arbitrator. In this way, the survivor is able to assume with reasonable confidence that its peer will be in a safe state after the watchdog expiry interval.
The practical difference between an arbitrator and a third node is that the arbitrator has a much lower footprint and can act as a tie-breaker for more than one cluster.
Option 3 - More Human Than Human
The final approach is for survivors to continue hosting whatever services they were already running, but not start any new ones until either the problem resolves itself (network heals, node reboots) or a human takes on the responsibility of manually confirming that the other side is dead.
Did I already mention you could add a third node? We test those a lot :-)
For the sake of argument, lets imagine I’ve convinced you the reader on the merits of a third node, we must now consider the physical arrangement of the nodes. If they are placed in (and obtain power from), the same rack, that too represents a SPoF and one that cannot be resolved by adding a second rack.
If this is surprising, consider what happens when the rack with two nodes fails and how the surviving node would differentiate between this case and a network failure.
The short answer is that it can’t and we’re back to having all the problems of the two-node case. Either the survivor:
ignores quorum and incorrectly tries to initiate recovery during network outages (whether fencing is able to complete is a different story and depends on whether PDU is involved and if they share power with any of the racks), or
respects quorum and unnecessarily shuts itself down when its peer fails
Either way, two racks is no better than one and the nodes must either be given independant supplies of power or be distributed accross three (or more depending on how many nodes you have) racks.
By this point the more risk averse readers might be thinking about disaster recovery. What happens when an asteroid hits the one datacenter with our three nodes distributed across three different racks? Obviously Bad Things(tm) but depending on your needs, adding a second datacenter might not be enough.
Done properly, a second datacenter gives you a (reasonably) up-to-date and consistent copy of your services and their data. However just like the two- node and two-rack scenarios, there is not enough information in the system to both maximise availability and prevent corruption (or diverging datasets). Even with three nodes (or racks), distributing them across only two datacenters leaves the system unable to reliably make the correct decision in the (now far more likely) event that the two sides cannot communicate.
Which is not to say that a two datacenters solution is never appropriate. It is not uncommon for companies to want a human in the loop before taking the extraordinary step of failing over to a backup datacenter. Just be aware that if you want automated failure, you’re either going to need a third datacenter in order for quorum to make sense (either directly or via an arbitrator) or find a way to reliably power fence an entire datacenter.
 Not everyone needs redundant power companies with independent transmission lines. Although the paranoia paid off for at least one customer when their monitoring detected a failing transformer. The customer was on the phone trying to warn the power company when it finally blew.