Their story of late has been pretty amazing. Going quite far in Europe but failing at home.
Summary:
2003 - 2004: Last 16 in the CL, but 2nd from bottom in the La Liga and relegated
2004 - 2005: 2nd in the Segunda, promoted to the La Liga
2005 - 2006: 6th in the La Liga, qualified for UEFA Cup
2006 - 2007: Last 16 in the UEFA Cup, but 18th in the La Liga and relegated
...
2010 - 2011: Promotion playoffs; lost after penalty shootout
2011 - 2012: Promoted to the La Liga
2012 - 2013: Survived relegation on the last day of the La Liga
From the Wiki:
Celta had a dramatic reversal of fortune in 2003—04. The previous season, they finished fourth in the league, putting them in the third qualifying round of the Champions League. Celta entered the group phase after eliminating Slavia Praha, and eventually reached the last 16 before being knocked out by Arsenal. However, their domestic form was disastrous, and they finished second to the last in La Liga, so they were relegated to the second level. Although their squad was heavily dismantled, they earned a return to the top flight at the first attempt, after finishing second in 2004-05's Segunda.
In the 2005–06 season, they finished sixth earning a return once more to the UEFA Cup. They made it to the last 16 in that competition as well, before losing to Werder Bremen.
In the 2006—07 season, Celta finished in 18th position and were relegated to Segunda División. At the end of June 2007, Celta avoided going into administration. However, if an agreement was not put in place between the club and its creditors within three months, then courts would declare the liquidation of the club's assets.
Due to the huge debt, the club was forced to sell many players and make tremendous cuts in the club's finances. Since then, they have been relying mainly in the reserves team, combined with some inexpensive signings. During the first three seasons in Segunda División the club struggled hard to avoid relegation, amid fears of the club's disappearance. This was a period of high instability, with constant changes of managers and players. In the season 2010–2011, the signings of the striker David Rodríguez, the winger Enrique de Lucas and the manager Paco Herrera turned the situation and the club finished sixth, performing a fantastic season and qualifying for the promotion. Nevertheless, they were eliminated in the first knockout by Granada CF on the penalty shootout, after an aggregate of 1–1 .