On an older/disused machine: the motherboard battery could be dead. Technically the RTC-NVRAM battery, commonly called "the CMOS battery". The standard part on virtually all computers is a CR2032 3.0V coin cell, available all over the place for less than a buck. Of course, replacing this component involves getting at the motherboard, which'll take a modestly technical, careful, and patient person somewhere around 15-60 minutes of disassembly/reassembly the first time through - it's the only battery in the machine (except for the big removable one, of course) and it's very clearly labelled. No guarantee (since I can't check your machine out myself, firsthand), but I'd say I'm about 80% confident this is your laptop's problem and the right fix; for a buck and maybe 15-30 minutes of patient effort you can't go far wrong.
I would then be inclined to think it's a main battery/power issue as well, but you say the battery appears to hold a full charge. Beware that if the laptop (and battery) are more than about 2-3 years old then it might report "100%" while in fact holding essentially no useful power at all, since lithium rechargables just don't last forever. Especially if they've been in storage for long periods.
If these solutions don't work then I would suspect hardware failure. Bad hardware failure like processor, motherboard, or RAM. You can (usually) get at the RAM quite easily, simply unseat/reseat it properly and quite often it'll work again. Other chips can "creep" out of their sockets, but I would think that you won't find that problem on any modern laptop. Could be hard drive failure as well, though that's unlikely since you'd get past the POST and witness all sorts of frustratingly cryptic error messages.
Final possibility: a BIOS password, the system is "locked" and simply won't progress past a certain boot point without receiving the right code. If so, then you can try several of the ridiculously common "backdoor" passwords, like "AMI", "AWARD", "DELL" - a google will find grand lists of these and you can simplify your search if you know anything about the particular manufacturers or models of BIOS, CPU, etc on your machine. If you're fortunate enough to have an Intel chipset and motherboard in the machine, then there's a very good chance that there will be a jumper (sometimes labelled, sometimes not) which can be configured to allow the machine to ignore this BIOS-password-level security while it's set.
I'm pretty sure that changing the cheap little CR2032 battery will fix this machine.
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Problems? See this thread at archive.org.
