You can do it with an oversized cabinet, but it won't be ideal IMO... you'll have to compromise on the size of several items (sump, skimmer, ATO reservior) to make it all fit. Not that you can't use adequate stuff, but a budget like the one you stated should build a top shelf system, and you don't have the room to go top shelf in a stand in my opinion.
And there is no way you'll be able to automate water changes given what you've said.
Don't get me wrong, you can get the necessary equipment in there to run a tank that size, but I'm going off of your intention to automate everything. You're still gonna be toting buckets!