Can I learn penetration testing at home?

Free options are few, but there are tons of videos and tutorials on specific attack vectors or products/tools. They will NOT make you a Penetration Tester, but they are free learning resources to learn penetration testing. Some decent options to start you off:

MetaSploit Unleashed: Learn an exploitation framework
SecurityTube: various videos covering a multitude of topics
NMap: The standard network enumeration tool
Web Application Hacker's Handbook: It's not free, but it is the bible on Web App Security
For practice, there are a number of resources:
Metasploitable VM (and other purposely vulnerable VMs)

• DVWA
  
• Mutillidae
  
• WebGoat
  
• Vulnhub
  
• hack.me
  

Do some searching on this site for other people offering opinions on free learning resources. But, the only way to learn is to get your hands dirty.