Can I collect points with just a single Reach RS unit? Or do I need to set up a base machine and rover?
Yes, you can collect points with just a single Reach RS unit. If you want to do this without paying for live RTK data then you will need to log data on each point for 5-15 minutes and then post process the data. The important part is whether or not you have access to such data. I don’t know what’s available where you live.
Google free CORS networks. This could give you live access to the data.
Here is an example of one network that supplies free data for post-processing. I use it with good results but it’s the only option where I live in Canada.
How do I know the accuracy?
That’s a good question. I’m not sure how to recommend confirm the accuracy of your fix but when you post-process you will see a collection of fixes that should all be located within a small block about 1 cm by 1 cm. Now if you only see one or two fixes I wouldn’t trust this but when you see hundreds of fixed points and they are all all within this 1 cm block you can have a high level of confidence in your positioning data. This is written as an amateur. Let’s see what the big boys say 
Good luck!