There are a total of 313 eligible batters this year.
Catchers: 41, including 3 who are eligible at one other position, and 7 who are catcher only (can't DH).
First base: 56, including 29 who are eligible at first base and at least one other position.
Second base: 56, including 35 who are eligible at second base and at least one other position.
Third base: 56, including 30 who are eligible at third base and at least one other position.
Shortstop: 51, including 23 who are eligible at shortstop and at least one other position.
Corner outfield: 133, including 25 who are eligible at corner outfield and at least one other position. (This includes all center fielders, who are eligible at corner outfield as well.)
Center field: 58, including 4 who are eligible at center field and at least one other position (in addition to corner outfield).
Designated hitter: Anyone can be a DH (except the 7 players who are eligible only as catchers). But there are five players who didn't play 10 games in the field at any position last year, and are only eligible to be used as a DH.
Remember, every team is required to field a team (one player for each position), plus a backup catcher.
Wait... only 41 catchers this year? Since we have 16 teams, and every team is required to carry two catchers, that only leaves at most nine catchers in the free agent pool! And some teams will draft more than two catchers. What if several catchers get injured at the same time and we run out of catchers (or players eligible at any other position)?

The Tom Princes for the other positions, where 250 plate appearances are required for eligibility:
1B: Luke Scott (.220 BA, .301 OBP, .402 SLG in 236 PA)
2B: John McDonald (.229 BA, .269 OBP, .308 SLG in 245 PA)
3B: Also McDonald, but if he's taken: Blake DeWitt (.265/.305/.413 in 243 PA)
SS: McDonald again, but if he's gone: Tsuyoshi Nishioka (.226/.278/.249 in 240 PA)
Corner OF: Shelley Duncan (.260 BA, .324 OBP, .484 SLG in 247 PA)
Center field: Bryan Petersen (.265 BA, .357 OBP, .387 SLG in 241 PA)
Here's another way to look at the position breakdown:
One position:
Catcher: 38, including 7 who are catcher only (can't DH)
First base: 27
Second base: 21
Third base: 26
Shortstop: 28
Corner outfield: 50
Center field: 54 (can also play corner outfield)
Two positions:
Catcher and first: 3
First and third: 2
Second and third: 6
First and center field: 2
First and corner outfield: 9
Second and corner outfield: 4
Second and shortstop: 8
Third and outfield: 2
Third and shortstop: 2
Shortstop and corner outfield: 1
Three positions:
First, second, and third: 4
First, second, and corner outfield: 2
First, third, and corner outfield: 2
First, third, and shortstop: 1
Second, third, and center field: 1
Second, third, and shortstop: 9
Second, shortstop, and corner outfield: 1
Third, short, and center field: 1
No position:
DH only: 5
And now... the pitchers.
There are a total of 382 eligible pitchers this year. There are 167 eligible starting pitchers; 203 eligible relief pitchers; and 12 guys who are eligible as swingmen (starter or reliever).
What if run out of pitchers? It's impossible. Even in a 16-team league, if each team drafted 10 starting pitchers, there would still be seven left in the pool. Every team could draft 12 relievers and we'd still have 11 left over. And that doesn't include the 12 swingmen.
By the way, the minimum league requirement is five starting pitchers and four relief pitchers per team.
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