To answer your question about $300 being enough to start, I would say no. You could do it on the residential side (I think) but most companies that will hire you will want some sort of insurance proof. For me, Thats about $600 per year. I guess you could find one allows you to pay installments though.
If not supplied (none of my clients supply cleaning equipment), you are going to need some sort of vaccum. Dont go to WalMart and buy a cheaper. You will lose creditibilty quickly. A good shoulder carried vaccum will run you about $350-$500. Look on Craigslist though for a used one. Dont forget Mops, Brooms, Mop Buckets, Chemicals, etc.
Here is the kicker on Commercial cleaning though. You wont get paid for almost 45 days. For example, say you start a service on Nov. 1. Most companies (not all but many) will want a bill at the end of the month. AFTER you dio the cleaning. Then, depending on terms, they wont pay you for 15 days after they receive the bill. (My terms are NET 15). So, is $300 bucks going to buy you all this stuff plus supplies for 45 days? Tough question. In this example you wouldnt get paid until Dec 15.
Hate to be a downer on this but wanted to give you an honest answer. With that being said, you can start a Residential Cleaning Company for $300. Best thing about that is they pay you the same day. What I did (and suggest you to do the same since money is tight) is to start on the Residential side then once you have a bankroll gradually move over to the commercial side.
It isnt easy to start any business and the cleaning industry is no different. Be persistent and hustle your tail off and you will be fine.
Hope this helps.