I've had position players with low health (30s), although not that low... at those levels of health, and running 20/20 training and medical, I assume a 15-day DL stint most seasons but don't expect long injuries or injuries with major ratings drops. If you can get those budgets up to 20 (also a good idea in general, especially for training) you should be able to get your money's worth out of that contract.
The same is not true of pitchers; pitchers with low health will typically hit the 60-day DL every few seasons.