This conduct is not lawful under US law
Under 17 USC 512 (c) (that part of the copyright law that implements the DMCA Safe Harbor provisions) this is probably infringement. 17 USC 512 (c) provides that:
(c) Information Residing on Systems or Networks at Direction of Users.—
(1) In general.—A service provider shall not be liable for monetary relief, or, except as provided in subsection (j), for injunctive or other equitable relief, for infringement of copyright by reason of the storage at the direction of a user of material that resides on a system or network controlled or operated by or for the service provider, if the service provider—
(A)(i) does not have actual knowledge that the material or an activity using the material on the system or network is infringing;
(ii) in the absence of such actual knowledge, is not aware of facts or circumstances from which infringing activity is apparent; or
(iii) upon obtaining such knowledge or awareness, acts expeditiously to remove, or disable access to, the material;
In this case the service provider either knows, or has good reason to suspect, That all or many of the uploaded files are game ROMs. and that these are uploaded in violation of copyright. The provider thus fails 17 USC 512 (c) (1) (A) (i) or (c) (1) (A) (ii) and has no safe harbor protection. 17 USC 50`1 provides that:
Anyone who violates any of the exclusive rights of the copyright owner as provided by sections 106 through 122 ... , is an infringer of the copyright ...
(b) The legal or beneficial owner of an exclusive right under a copyright is entitled, subject to the requirements of section 411, to institute an action for any infringement of that particular right committed while he or she is the owner of it.
Under sections 502, 503, and 504 the site owner would be subject to a suit with the possibility of injunction, actual damages, statutory damages, and/or costs and attorney's fees. Under section 506 criminal prosecution is possible (although not likely in the case described). One of the right protected by section 106 is that of making a copy of a protected work. By allowing an upload the service provider is at least assisting in making a copy, and is storing that copy. The service provider is thus an infringer, without the protestation of the Safe Harbor provisions.
That only the uploader can access the files may reduce the damages, but will not change the fact of infringement. And as a practical matter, a user willing to share passwords and other security info can use such a service to distribute this content. Such distribution would further infringe copyright.
There is no basis for fair uase to apply.
Tye US fair use concept is spellesd out in 17 USC 107
This says use is for:
purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research
"Backup" is not mentioned.
The factors to be considers for a fair use determination are:
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
If the service charges users (1) tilts away from fair use, otherwise not. The RONs are creative, and thus (2) leans away from fair use. The entire ROM is used, and thus (3) is against fair use. Providing access to ROMs harms a potential market for them, albeit one the owner is not currently exploiting, and thus (4) leans away from fair use.
A fair use defense seems pretty sure to fail.
A service as described in the question would, under US law, be an infringement of copyright, and the service provider would potentially, at least, be subject to a suit with a possibly significant damage award should any of the copyright holders notice and bring suit. There is a fair chance that no one would ever act on this, but that is not any legal defense.