Is it possible for both sides of an encounter to be surprised and if so what would happen? This is a hypothetical idea I just came up with but it made me wonder what would happen. Would the surprise for all intents and purposes cancel each other out as nobody can act on their first turn or is there some other way for things to play out. If you can provide an example in your answer so I can better understand it.

Surprise is very much up to the GM

How does surprise work?

The section on "The Order of Combat" states:

COMBAT STEP-BY-STEP

1. Determine surprise. The DM determines whether anyone involved in the combat encounter is surprised [...]

The section on "Surprise" states:

The DM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. Otherwise, the DM compares the Dexterity (Stealth) checks of anyone hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn't notice a threat is surprised at the start of the encounter.

One could interpret this to mean that the only way to be surprised is when a creature makes a stealth check. Then, if their stealth check exceeds your passive perception and you don't notice their attack/threat, you are surprised.

That said, in my opinion, stealth is an explanation of one way to determine surprise; ultimately the last sentence is the rule that a GM can apply when determining surprise, regardless of stealth checks:

Any character or monster that doesn't notice a threat is surprised at the start of the encounter

For further evidence that this is the case see these answers: "How does burrowing interact with surprise and attacks from hiding?", "In this scenario, how do I determine whether the enemies are surprised?", and "Can a PC be surprised if he expects combat before the encounter?". For evidence to the contrary see the answer here "Should players get a surprise attack by interrupting a villain's speech?".

Thus whether this happens is up to a GM and/or the stealthiness of the sides.

Can both sides be surprised?

Perhaps it is not possible for the entirety of two sides to be surprised, this would require both sides to view the other as a non-threat and yet wish to have combat happen regardless. What certainly is possible, is for all but one character to be surprised, such as if they initiate a combat where nobody expected was going to occur. Everybody would be surprised except the initiator.

A possible way mentioned by user @Ryan Thompson is if one party triggered a trap and were teleported into a room where another party is not expecting their arrival.

What happens if both sides are surprised, even if it may not be possible?

If both sides somehow are surprised the following would apply to both sides:

If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. A member of a group can be surprised even if the other members aren't.

Thus the first round of combat would be everybody not moving and not taking actions. Additionally reactions couldn't be made until after their turn had passed but as nobody is doing anything, they can't actually react to anything.

Notable exceptions occur if only some of a certain group is surprised, perhaps one of them has the Alert feat, or is under the effects of the foresight spell. Another time this may matter is if there were casters who were in the middle of casting long casting time spells. The result in this specific scenario is debated in the question "What happens if a caster is surprised while casting a spell with a long casting time?"

No.

An encounter does not start, initiative is not rolled, and surprise is not determined, until some combatant has reason to attempt to initiate it.

For all combatants to be surprised, that must mean that no combatant has noticed a threat.

And if no combatant has noticed a threat, why is initiative being rolled?

There is an important distinction to be drawn between surprised - the english word describing an unexpected occurrance - and surprised - the game mechanic by which some combatants cannot act during the first round of an encounter.

It is absolutely possible for everyone involved to be surprised, because they did not expect an encounter to be occurring. Such could happen when a party falls down a pit into the middle of a group of enemies - certainly the enemies are surprised to see adventurers so suddenly, and certainly the adventurers are surprised to land in the midst of a bunch of monsters, but it is not possible to claim that either side "did not notice a threat" - so neither side should be surprised when initiative is rolled.