Can I set up a website without a domain name?

and is it possible to design it at home?

Most Internet Service Providers (ISP’s) provide a “personal webpage” option for their clients. Depending on who you get your internet from this may be an option. There are also other sites on the net that will give you space in exchange for having advertising on your site. These services are only really appropriate for non-corporate websites.

It is also possible to design and create a website from home - however only a very, very, basic one.

You could also try searching google for stock web-site templates -but again- only if your needs are very basic.

If you are intending to produce a website for anything commercial in nature I would invest in a domain name (12$) a basic hosting package (20$ per month) and a web designer. (less $ than you might think)

The above rates are from these guys: