So - your use of terminology would lead me to think that you haven't been at this too long (I apologize in advance for the snark if that is not the case). If you deal with certain information, you would certainly NOT use the term CLASSIFIED in discussing the status of that information. CLASSIFIED has a VERY specific meaning in certain domains - including the domain that you seem to indicate that you work in. If you are, indeed, handling such information, I would suggest running, not walking to your FSO for a conversation. It will probably be fairly brutish and short.
If, however, you are dealing with ITAR regulated information, then you have a different set of issues. You may not export the data without a permit, but you don't need to control it specifically within the US. Also, the regulations around foreign persons (or those of dual nationalities) relate to export activities. So, you can't transfer to a foreign person if you know (or suspect) that they are going to export the data. However, foreign persons in the US that aren't an export channel are not an issue (else a whole lot of commerce in the US would halt since I have no idea if another company has any foreign nationals employed, and I don't have to get an ITAR export license to ship something to another domestic company).
In the later case (where we are talking regulation, not classification), you don't have an issue if you don't export the data (don't pick a company with foreign presence for cloud storage). Actually, one could probably be ok if they encrypted it (strongly) and then stored (but you may (or may not) want to talk to your DDTC rep about that.
You should have no problems finding an offsite storage company to provide the service, and/or use someone who allows you to restrict the S3 zones (if AWS is the backend store) to us-* regions. Similar for rackfiles, dream objects, etc.
Another comment here is worth highlighting, however - use consumer services, get consumer service. Go upmarket a bit if you are actually looking for something that your company's bottom line is hung on.