Hello, I was recently admitted to UCSB which is a dream school of mine and I was extremely excited when I got accepted. The biggest downside is that I wasn’t admitted to my first or second major choice (Mechanical and Chemical Engineering). I contacted the school and they can’t offer me a spot for the Engineering department, but they are willing to offer me a spot in the College of Letters and Science, so this leads to my questions. What is major is the closest to Mechanical Engineering or what major can help me get a job as a Mechanical Engineer with a non-engineer degree? Is it even possible or am I screwed? 
It’s not really possible to become an engineer without an engineering degree - engineers require a very specific educational background. However, it is possible for you to potentially complete another major and then return to school for an MS in engineering, if you plan carefully.
You should consult with an adviser (preferably a professor in the engineering department) to really plan this well, and look at the requirements of a couple mechE MS programs. However, a good move is probably to major in physics, taking a good amount of coursework in mathematics as well and as much in the engineering department/school as you can.
Paging @xraymancs, who may also have some good advice for you.
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Some physics graduates do go into engineering jobs ( https://www.aps.org/careers/statistics/bsprivatesec.cfm ), but that is not a desirable path to take now if your goal is to work in engineering, since graduates with the appropriate engineering degree will be the first choice of employers. In particular, physics does not include the engineering design courses and projects that prepare the student for the design aspect of engineering jobs.
If some other affordable school admitted you to your desired major, or is not competitive to get into your desired major, it would be a better choice than UCSB, where getting into engineering majors is highly competitive ( https://engineering.ucsb.edu/undergraduate/academic-advising/change-major-college-engineering ).
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I agree with @ucbalumnus about choosing a university which has admitted you for engineering if you have that option. Many universities have strong, ABET accredited engineering programs and you can get an excellent education on a par with UCSB there.
If UCSB is your only alternative or if that is where you want to go for other reasons, then @juillet is correct that your best bet is physics. You might be able to get some design courses as electives to gain the background that you need for a job in engineering.
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