How can I start a business while on my F-1 and applying for EB-5?

Complicated decisions, and on an F-1, you could set up a business, but you can’t work in the business in any way without a work visa. Potentially, if you filed the EB-5 concurrently with the I-485, you would obtain an EAD to be able to work through that process.

Generally, you can pursue both, but the timing and structure matter because F-1 status has strict work restrictions, while EB-5 is an immigrant-investor path with different requirements.
While on F-1, you generally cannot actively run a business without employment authorization (EAD) through having been granted Optional Practical Training. There may be other nonimmigrant visa options that would allow you to work, but that would require analysis of your background. As an F-1 student, you may usually own or invest in a company, form an LLC or corporation but primarily this is based on your being a passive shareholder or co-founder and NOT working.
On the other hand, the EB-5 is a totally different vehicle to ultimately allow you to qualify for a green card, with the potential that you could file for your adjustment of status and with that application also apply for an Employment Authorization Card (EAD) through the adjustment of status filing. The EB-5 green card case is based on making a qualifying investment that creates U.S. jobs. In most cases, a startup will not automatically qualify for you to obtain permanent residence (green card status) unless you've made a sufficient capital investment, can provide evidence of the lawful source of funds and that as part of the EB5 filing, you present a comprehensive business plan that demonstrates the creation of at least 10 full-time jobs for U.S. workers.
Generally, I would counsel clients that hold F-1 status to tread very carefully with respect to forming a business unless they already have employment authorization or are otherwise satisfied with being a passive investor with no actual responsibilities that could be construed as being a violation of the F-1 student visa status.
I hope this helps.

The choice regarding selecting visa programs and/or opportunities is often based on what is authorized under one’s status, and how one’s long-term plans play into them. Ultimately, the timing to secure benefits or access becomes the most important among the relevant things to review. For example, some inquiries that might matter include: Does getting OPT help my plans? Is applying for EB5 early and securing work authorization important? Where and how do my travel needs impact my other decisions?
There are, of course, many other potential questions. These are often very personal, and there is definitely more than one valid approach.