I mean allowing users to access movies and stuff from their Mac...
That's largely constrained by your available Internet network bandwidth.
In the "cloud".
I've heard that term to mean...
"Cloud" = "client-server computing"
"Cloud" = "Service Bureau"
"Cloud" = "JSON or AJAX"
"Cloud" = "Co-Lo"
"Cloud" = "Amazon-like Hosted Services"
Again... The term is largely and unfortunately meaningless in a discussion, because everybody has their own definition of what it means (to them). The marketeers will particularly prefer to surround it with what can be rather magical thinking, unfortunately.
I need the server anyway to power the network side of my iOS apps, but I'd much rather have the Mac Mini Server used for everything it offers instead of just having a MySQL database and running simple PHP scripts.
It'll work for that. Co-lo is a good option here, too, as that can avoid network bandwidth constraints.
Stuff like typical mail services and iChat text chat don't need much bandwidth, either.
If you're hosting small stuff for a few folks and maybe one or two dinky-window video streams, then a typical DSL connection can work. If you're hosting a whole lot of video (as is common with some definitions of "cloud" services), then you'll tend to slam into the network pipeline. A Mac Mini Server can run three or four full-HD video streams over Gigabit Ethernet reliably, but'll hit the wall if you tried that same transfer over a DSL link. (And with DSL (ADSL), the uplink bandwidth can be low, and sometimes the associated latency can be large.
I've had a single iChat video session and a parallel screen sharing session get really choppy on a mid-grade DSL line, too.