Gmail and Google app’s has a space limit for it’s free service. Most of the time people don’t delete emails

You can search for a load of emails via : 

“after:2011/1/1 before:2011/12/31”

Once this is done you can tick select all tick box and delete all then delete.

Remember to delete these from your Trash as well

There is also a script to speed this up here 

https://github.com/pariswells/pariswells.com/blob/master/bulkdeletegmailitems.script