You're right, you or some peer needs to keep it online. That could be a service, or using the hyp cli tool, or beaker itself, or etc. There's gotta be one peer somewhere.
I don't know the Hypercore community well enough to be able to say for them, but I know it's similar with IPFS, and for it there are already a few commercial services providing such kind of hosting/replication (a.k.a. "pinning"). During a recent hackathon at my workplace, I led an experimental project based on such services:
I'm particularly excited about the prospects of setting up a peer of my dat archives on my mobile phone. It's an always on, power efficient computing device. Not ideal for speed, but seems like a great fit for a last-ditch-effort layer of redundancy when hosting on hypercore.
There used to be a service/project called Hashbase[1] that would "pin" (aka seed) a site for you. Looks like it's inactive, though, which is a shame since it feels like something along these lines would be pretty necessary for any Beaker sites you might consider remotely real/serious.
EDIT: don't talk about up/down votes, my bad