Can you have a career in academia without having to write grant applications? I really like doing research, but I've heard that all the fights for funding can be a major headache. So I am wondering, how can I enjoy my work without worrying about grant applications? Are there specific fields or positions that don't require this?
The need for grant funding depends a lot on the type of research that you do.  The more that your research requires laboratory facilities and equipment, graduate student research assistants to run the equipment, or travel for field work than the more that you're going to need grant funding.  In the other direction, grant funding is less important if you do more theoretical research by yourself or with collaborators at other institutions.  

No matter what type of research you do, you're going to need time to work on the research and some money to travel to conferences and present your research.  

One of the few ways in which this can be done is to hold a faculty position at an undergraduate liberal arts college.  Some of these positions have reasonably light teaching loads (although typically higher than at research universities) and funds to send faculty to conferences.