Memory full..

KAC1990​  No, there is no way to play with content that's not installed locally on your computer.  I mean, you could subscribe to a streaming service where none of the content is installed on your computer, because it's installed on a computer or server the company owns, but that doesn't sound like what you're looking for here.

However, is the issue memory or storage?  Memory refers to RAM, the working memory of the computer, where it puts data from apps that are currently open.  Storage is your hard drive or SSD, which contains all the data you've created or downloaded whether you're using it or not.  Installing Sims 4 takes up storage; running it takes up RAM, but the amount you need for all packs is not always all that much more than what you need for the base game and just a few expansions.

If you're not sure what the issue is here, please post a screenshot of any error message you see.  Please also provide a dxdiag, for reference.  Click Windows key-R, enter dxdiag in the run box, wait for the scan to finish, click "Save all information," and save the file to your desktop.  From there, you can attach it to a reply using the paper clip (Attachment) icon included with the other formatting buttons.