You may not know that according to:
http://support.microsoft.com/kb/2764086/en-us
that, amongst other things, SharePoint Foundation 2013 is not supported on a stand alone workgroup (non domain joined) server.
Now just because it isn’t supported doesn’t mean that it can’t be done. Well almost. Here’s what I’ve worked out so far to get it almost all the way. I did the following on a Server 2012 that was not domain joined, i.e. only in a workgroup. You’ll also need a version of SQL 2008 or better installed before hand.

So you start the install and get the splash screen above. Select the option to install the pre-requisites.

Next you’ll get information about what will be installed. Press Next to continue.

Accept the license agreement and press Next.

The required software will be downloaded and installed.

When that is all complete you’ll get a summary and when you press Next the server will be automatically rebooted.

After you log back in the installation will continue installed some more software.

When complete you’ll see the above message. Best bet here is to do another reboot to ensure all the installations are complete.

When the reboot is complete re-run the installer and you will see the splash screen again. This time select the option to Install SharePoint Foundation.

Accept the terms again and press Continue.

Select the Complete option and the Install Now button.

You’ll need to wait while more software is installed.

When complete uncheck the option to Run the SharePoint Products Configuration Wizard now and press Close.

Locate the SharePoint 2013 PowerShell console and at the prompt run:
new-spconfigurationdatabase
Enter the details for your configuration database name, database server, farm credentials and finally a farm passphrase (and don’t forget this!). This process will now create the configuration database for you and may take several minutes to complete.

When the PowerShell command has completed locate and run the SharePoint Configuration Wizard. Select the option to not disconnect from the server farm.

Set the Central Administration Web details to what you require.

View the summary.

The configuration wizard will complete.

You should now get to the point of seeing the SharePoint 2013 Central Administration site. although not generally a good idea select the option to Run the Wizard to attempt to configure everything.

Select the Service Account (here I used the administrator which is not best security practices).

If you scroll down you’ll see all the option checked except for the Lotus Notes Connector. Continue.

Here’s where I get the error with the Search Service Application. Press Next to continue anyway.

The wizard will complete and you’ll have a SharePoint site like so:

Problem is the search service won’t be running which obviously has something to do with that error mentioned previously.
Apart from that you should have a a SharePoint Foundation 2013 site running on a stand alone server. Neat. Again, remember that is NOT a supported configuration.
I’m sure I can find a solution to the error given some time so watch out for a future update on the topic.