The rules for mounting a creature say "A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount." So a Medium-sized creature, like a Human, can ride on a willing creature that is Large-sized or bigger.
A Mule is Medium-sized, but has a "Beast of Burden" ability: "The mule is considered to be a Large animal for the purpose of determining its carrying capacity."
Does this counting as Large for "carrying capacity" also mean that it counts as large for purposes of being ridden as a mount? It sure seems that logically a human ought to be able to ride a Mule, and I may allow it in games I DM simply on that basis, but I'm not sure where in the rules it's explicitly said.
I think a big part of my confusion is that carrying capacity seems to be defined in the list of mounts in the Equipment chapter. There it says that a mule's carrying capacity is 420 lb., and creature carrying capacity doesn't seem to be related to creature size at all. So I'm not sure what the "Beast of Burden" ability is supposed to mean at all.
Best Answer
A medium character can probably ride a mule awkwardly, but only a small or tiny character can effectively use a mule as a mount in combat.
The Beast of Burden ability explicitly states exactly what game mechanic it affects - how much weight the mule can carry. It doesn't have any effects beyond that. The mule is strong enough to carry most medium-sized creatures, and that's probably good enough for transportation purposes, but it's just too small for them to be able to effectively ride it as a mount in combat.
Your confusion will hopefully be resolved by looking back at the Ability Scores section of the basic rules, which is where the rules for determining carrying capacity are described:
The mule, as medium creature with a strength of 14, would normally have a carrying capacity of 210lbs - but because its special ability means it is treated as large for the purpose of determining carrying capacity, that value is doubled to 420lbs (the value given for the mule in the mounts table).
Not all mules are equal and not all mules are Mules
It may be worth keeping in mind that horses/donkeys/mules are creatures with considerable variation of size between individuals/breed - wikipedia suggests that weight can range between a mere 50lbs to over 1000lbs. One creature stat block will never accurately describe all of them (you may note there are several different varieties of horse to choose from in the creature lists, including the much smaller pony). Whilst we can easily imagine a mule big enough to serve as a combat mount for a human, and such mules undoubtedly exist in the real world - that's not the mule as described by the Mule creature stat block, which presumes a relatively small equine. A larger mule should probably be represented using the statistics for an equivalent horse - the fact that it is actually a mule is not important for the mechanics of the game.