Wade Boggs and Tony Gwynn always seemed like legitimate threats to hit .400 (Gwynn might have actually done it if not for the '94 strike), so it makes sense that both players accomplished the feat over the course of 365-day periods:
Wade Boggs (June 9, 1985 to June 6, 1986)
Tony Gwynn (July 3, 1993 to July 2, 1994)
Boggs and Gwynn are deservedly in the Hall of Fame. Another phenomenal hitter and all-around great player who should be in the Hall of Fame, Larry Walker also flirted with .400 on occasion. Unlike Boggs or Gwynn, Walker was a power hitter. Not a natural .340-hitter like those other two, his numbers were boosted by playing his home games at Coors Field in the late 90s. But oh, the numbers he put up...
Larry Walker (July 19, 1998 to July 18, 1999)
And finally, Mark McGwire's 1999 campaign overall was a slight fall-off from his spectacular 1998 season. After a slow start, Big Mac was hitting .225 with 8 homeruns on May 19th. But from that point on he was better than ever, going on an unprecedented rampage that really only subsided when injuries slowed him down in 2000.
Mark McGwire (May 19, 1999 to May 18, 2000)
For this 365-day period, McGwire had more RBI than hits, and almost half of his 149 hits (and .302 batting average!) were homers. He finally spent an extended stint on the DL in July and August, and retired following a disappointing 2001 season.