One of the earlier Imgur style hosts, ImageVenue, did this very successfully (they were extremely large in terms of traffic at their peak and for the time, until better options came along).

The founder, Vlad, used to rent out cheap dedicated boxes with about 500gb of bandwidth per box for $60 to $80 per month. He'd cycle upload machines - eg 123.imagevenue.com - and fill up the drives on them until no more space was available; when bandwidth was exhausted, that was that for the month. Typically image file downloads would decline on any given upload over time, so machine resources would free back up in terms of bandwidth and deleting images that weren't viewed after N amount of time.

For image-heavy use I found its just cheaper to build your own box and cohost instead of renting.

How do you figure that? You can rent the exact same server as a dedicated server. The difference is, you have no capital outlay up front, you don't need to ship and rack it or replace broken parts and there is only a month-to-month commitment.

The boxes with lots of disk and memory are much more expensive when rented vs buying. SSDs are super expensive to rent, for example.

I made a Skylake Xeon system with all enterprise components for $2k. You can't even rent Skylake Xeons anywhere right now.

I've never had problems ordering custom hardware from my dedicated server providers. At worst, all I've had to do is pay a one-time fee for a particular component. SSDs have never been a problem. Neither has any Intel CPU.

It of course depends on your provider.

Rent vs buy depends a lot on the circumstances. Maybe if your requirements don't change and you don't upgrade your server hardware it's cheaper in the long run to buy. However, when I've kept a server for more than a year, I've always been able to negotiate a discount.

You didn't post your config, so I can't tell how exotic your requirements are.