This depends on company policy and practical considerations.
If you can leave your computer running overnight or over the weekend, great, that's an easy solution. This could create security issues: If you leave your computer logged on, what if another employee, or someone from the cleaning crew, or some other unauthorized person, comes along and uses your computer to access sensitive information or vandalize valuable data?
If security is a problem, is there a locked room where you could leave the computer while it runs? Or a computer that is in a secure place that you could use?
Are you allowed to take a laptop home? If so, you could take it home and let it run overnight there. Presumably you don't have problems with strangers wandering around your house, so security is unlikely to be an issue. A download doesn't normally require you to keep constant watch. You start it going and come back in a few hours to see if it's done. So it's not like you'd be giving up a lot of your personal time.
Is there someone else who can keep an eye on the computer while it runs? Like if there's a day shift and a night shift, can you start the download and then before you leave for the day ask someone else to watch for it to finish, and then shut down your computer? But most IT departments don't have multiple shifts, so this is iffy.
Any way to break the download into pieces, so you can download part today and part tomorrow?
You said your contract doesn't provide for overtime. Can you re-arrange your schedule, like work 10 hours today and 6 hours tomorrow?
In the final analysis, I'd talk to my boss about it and see what he considers an acceptable option.