No Ground, no way
Theres no way I would run without a ground. Even using a power strip will offer little help. No ground then no place to dump that extra power. It only takes one surge or lighting strike on the system. If you can't get it out a window or other opening I would use a 14 to 12 Ga. solid wire along a baseboard and to the closest water pipe. They make clamps just for that. If you use an adaptor, I would make sure it has a real good connection and not just wrapped around an eyelet.
I wired in Isolated Ground outlets for all my electronics. Each outlet has a separate ground to a 8 foot grounding rod. I use APC backup units for my comps. We get a lot of lighting storms here and 20+ years I've never had a problem with comps. I can't always get to them to shut them off in time, otherwise three run 24/7