Have mac sierra 11.10.5. Need to upgrade to 11.11. Machine can't take 11.13. How can I upload 11.11

Upgrade to OS X El Capitan - https://support.apple.com/HT206886 - "OS X El Capitan remains available for Mac computers that can't upgrade to macOS Sierra, or that need to upgrade to El Capitan first."

No such thing as 11.10 or 11.11. Currently the highest macOs version is 10.13.3 High Sierra.

Sierra was 10.12.x

What model MacBook do you have exactly? If its running Sierra, it can run High Sierra.

Identify your MacBook - Apple Support

If its running 10.11.x El Capitan it may or may not be able to run Sierra or High Sierra.

macOS Versions Names and Numbers - Wikipedia.org

I believe OSX 10.11 is El Capitan, not Sierra. I think since there seems to be some confusion here it would help to know exactly which model Mac you own (use the link that brought you here) and to what version you wish to update/upgrade (if that is even possible).

You can get El Capitan from the App Store if your Mac cannot update to High Sierra as explained in the link provided by Limnos above.