Eligibility for the Hall of Fame is two fold. There are automatic nominations that the AI processes and these players are automatically eligible to be voted into the Hall of Fame. There are also user nominations that occur at the beginning of the season and last through the first day of Spring Training.
A player will be eligible for the Hall of Fame as soon as he retires and fits the specified criteria outlined below. In other words, there is no 5 year waiting period like in real life. The team that he is inducted with is the team the he plays the most games in (Pinch hitting appearances do not count as games played).
Once nominated for the HOF a player has five seasons to be voted in. After that they will no longer be eligible.
Auto-nominations - Position Players: 7+ Silver Slugger, 10+ Gold Gloves, 7+ All Star Appearances, 3+ MVPs, 500+ HRs, or 3,000+ Hs. Pitchers: 7+ All Star Appearances, 3+ Fireman of the Year, 3+ Cy Young Awards, 300+ wins, 3000+ Ks, or 500+ SVs
User nominations (regular season stats only) - Position players: 10+ years of ML experience with 300+ PA per season. Relief pitchers: 10+ years of ML experience with 45+ G per season. Starting pitchers: 10+ years of ML experience with 125+ IP per season.