It is possible to learn violin without a teacher, but you will almost certainly pick up bad habits, and probably severely limit your progress when you do things wrong and it takes you several months or years to figure out what a good teacher would know in 30 seconds. This is true whether you've never played an instrument before or you're a skilled musician on other instruments.
I'd strongly recommend getting a teacher for at least a couple months at the start. The violin is very difficult to get a good sound from, or to play in tune on. A teacher can drastically cut down your frustration at the beginning, and once you can play Twinkle, Twinkle Little Star in tune, you've at least got enough basics to play music instead of screeches and wails.
If the only reason you don't want to take lessons is that you have an irregular, busy schedule, look for a teacher who can be flexible about when you have lessons week to week. Independent musicians who teach are often more flexible than music schools with a central scheduling policy. When I've done this (with 4 teachers and 2 instruments), the musicians work me into places where kids have canceled, or put me at the end of a day so they don't have gaps in their schedule. I have the flexibility to work lessons around my schedule, rather than the reverse. The drawback is that some weeks, they might not have an opening that works for me, but it usually works out.
If you go the teacherless route, look for opportunities to play with others, even if you don't think you're ready. Find workshops and festivals you can do on weekends, and look for a local open jam or no-audition orchestra. By playing with better players, you'll get advice and see what other people are doing. It's also good motivation.