I’ve done exactly this. Recently I turned on a computer after 25 years. I plugged it in, turned it on and nothing happened. I left it plugged in for an hour to let the battery get a little charge, tried again to turn it on, and it worked fine, although the computer was so old that there was no Ethernet or Wi-Fi to connect to the internet.
Old systems that can connect to the internet are desperately susceptible to attacks because they are unpatched. They should always be connected behind a firewall or they can be be attacked within a few seconds to a minute.
Old systems often can’t run modern browsers and so won’t properly display modern web sites.
The most typical failures on these old systems are the motherboard batteries and the hard drives. If the battery is dead, it might just cause the date to be wrong or it might prevent the CPU or DRAM timings from being configured. In other words, the result might be unnoticeable or it might prevent the system from working at all.
A bad hard drive can result on an “Operating system not found error” or it might cause a hang or crash while running.
The most interesting part of starting old computers from my family is the snapshot in time that it captures. The documents from Microsoft Word, the browser history, the photos, the old emails. It’s an interesting trip down memory lane.
Update (7/2024) Many people have said that there’s a very low likelihood that an old computer will be attacked. Watch this YouTube video, where an old Windows XP machine is attacked and infected in just ten minutes! (Which is more than the “few seconds to a minute” I mentioned above, but if a random scan finds one machine in ten minutes, it will certainly find some machines much faster.)