What are some simple sufficient conditions for the existence of a uniform distribution on a set?
Specifically, what are some conditions on the set that aren't much more complicated than what you might see in undergraduate analysis (compact, connected, etc) that might guarantee the existence of a uniform distribution?
I am not sure if this is what you are looking for, but I'll give it a try. Assume that you have a Borel set $B \subseteq \Bbb R$ such that $B$ has finite and positive Lebesgue measure, i.e. $\lambda(B)\in (0, \infty)$. Then the normed and restricted measure $$ \mu: \mathcal B \to [0,1]: A \mapsto \frac 1{\lambda(B)}\lambda(A\cap B) $$ is a uniform distribution on $B$.
On the other hand, you can define a discrete uniform distribution on every finite set $\{x_1,\dots,x_n\} \subset \Bbb R$.
Moreover, there is no uniform distribution on unbounded sets.