How to repair backup of .mdf file database?
You're getting this while trying to backup any database?

msdb is used for a bunch of things and you'll want to make sure it's okay. But critically, it sounds like your corruption is in a table which is used for backups, and this is a good thing, because you don't necessarily need that data.

Start by scripting out all your SQL Agent jobs, exporting all your SSIS packages that you've put in msdb, and anything else you can think of that uses msdb.

Now do some investigation to find out what that corrupt page is used for....

...but ultimately, your best bet is likely to be to stop the instance for a moment, pull out the msdb MDF and LDF files (so you have copies), restore from a backup from a different database (and check again for corruption), and then redeploy all your maintenance jobs, agent jobs, SSIS packages, and anything else you can think of. 

You might be able to swap your MDF and LDF files onto another machine and extract extra information that you may have missed, but this should at least get you going again.