If there is a work that is out of copyright (was published in the 1800s) but I can only find it in an online repository, can the repository claim a copyright or other limitation on my use of their scanned copy?
In the US, the original author still owns the copyright, unless the original work was out of copyright (in which case no one owns the copyright): copyright is not granted for just pressing a "scan" button, and mechanical reproductions are not eligible for copyright (copyright protects creativity, and a scan involves zero creativity). To quote 313.4(A) ("Mere Copies") of the Compendium of US Copyright Office Practices, Third Edition:
A work that is a mere copy of another work of authorship is not copyrightable. The Office cannot register a work that has been merely copied from another work of authorship without any additional authorship. [...]
- Photocopying, scanning, or digitizing a literary work.
That doesn't necessarily mean the repository can't put any restriction on your use of the copy; the repository may have a terms of service. Terms of service are rooted in contract law, not copyright; just because the work is in the public domain, doesn't force the repository to show it to anyone who asks, and they can force you to agree to a contract first. The enforceability of that contract is fact-dependent.