The PhB states
You can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you.
If a PC has 4 enemies surrounding him on the north, south, west and east edges, is that PC unable to move?
The distribution would be something like this:
O A O
A B A
O A O
O is an empty space, A is an NPC and B is the PC.
This PC can move diagonally. Doing so does not involve moving through a hostile creature's square.
(See PHB p.192: Variant: Playing on a Grid)
Characters (and monsters) can move diagonally like they do horizontally and vertically, as a one-square move. From the Entering a Square section we can tell that moving into a diagonal square doesn't involve moving into either of the adjacent squares that border it: with only 5' of movement left you can enter the diagonal square.
Diagonal movements can be restricted by an obstruction that completely fills the adjacent-not-diagonal squares—see the Corners section. But that section specifically calls out terrain and trees—stationary objects, in other words—as the obstruction. A medium-sized creature does not create this type of obstruction: as described on the previous page (Space), the opponent doesn't fill the square it's in.
There is no recognition of the longer path between two diagonal centers than between two truly adjacent centers. The PC can move to any of the corner-spots as one 5' portion of their movement.
The DMG has alternate rules for counting diagonals as alternately 5' and 10' (p. 252), but this only matters for distance considerations—even under that "Optional Rule: Diagonals" your PC could still move to a diagonal space.
Beware opportunity attacks, though!
From the movement rules in the PHB:
Corners. Diagonal movement can’t cross the corner of a wall, large tree, or other terrain feature that fills its space.
The space of another creature, whether hostile or not, also counts as difficult terrain.
you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you.
A creature's space is the area in feet that it effectively controls in combat
A creature’s space is a terrain feature that fills its space. Therefore it can’t be crossed diagonally.
In my opinion, the most voted answer is contradicted by the PHB and thus incorrect. The RAW answer to your question is NO if using the Variant: Playing On A Grid (PHB 192) rules. You cannot move diagonally in that specific scenario illustrated in the question. In the PHB page 192, it states
Corners. Diagonal movement can't cross the corner of a wall, large tree, or other terrain feature that fills its space.
On page 190 of the PHB under the heading Difficult Terrain, it states
The space of another creature, whether hostile or not, also counts as difficult terrain.
And then on page 191 of the PHB under the heading Moving Around Other Creatures, it state
Remember that another creature's space is difficult terrain for you.
Both of these statements imply that a creature is terrain and serves as a corner for purposes of diagonal movement.
PHB pg 192 actually says diagonal movement can't cross the corner of something that fills its space. This has a completely different meaning than if you read it as if it says "doesn't" as nitsua claims and it follows more closely with older editions.
To say that something cannot do something places a restriction on it in that it would be breaking the rules if it happened. Saying that something does not do something eliminates it from the set of considerations.
