There are dozens of truly awful teams to choose from. If we exclude amateur entries intended to take part in their home GP only, and also teams which didn't build their own cars, then some of the worst ones to have started a race are:
1950s: Plenty of amateur efforts, but Alta stands out for the Irishman Joe Kelly actually managing to be unclassified in successive British GPs, 13 laps behind in 1950 and 15 laps down in 1951...
1960s: The stand-out worst team for me is A-T-S (1963-64). The car failed to qualify more times than it started, and the engine could only be removed from the chassis by using a hacksaw. Never qualified in the top 10, never scored a point.
1970s: A hard choice. But I'd go for Merzario (1978-80). Failed to qualify twice as often as it started, and never higher than 20th. Not one single race finish. The cars were bulky, ugly things which never had the basic speed to develop further.
1980s: Another decade with a lot of choice. Despite the prescence of Eurobrun, I think Coloni (1987-89) has to be the worst. More often than not, the car failed to prequalify. Half the rest of the time, it failed to qualify. And on the rare occasions that it did race, it broke. One 8th place, where some very late retirements moved the car up several places, was the highlight.
1990s: There were a few weak teams at the start of the decade, but nothing (not even the antiquated Forti) was as bad as the Andrea Moda (1992). Failure to pay the entry fee. Not having the cars ready. Drivers not having a superlicence. Forgetting about scrutineering. There were many way in which Andrea Moda contrived to make their slow car even less competitive. The car qualified just once, 26th and last at Monaco, and retired from last place when the engine broke.
2000s: The quality of teams has increased in recent times. Only HRT really qualifies for the worst team list, and even then they wouldn't have been THAT bad in past times.
Of course, for me there is only one contender if we include the teams that never even managed to qualify. Life entered just 14 races, but failed to even get out of prequalifying once. Gary Brabham made his bid for the worst qualifying performance in history: 143% of the pole time in Phoenix. He was replaced by the veteran Giacomelli, who had no greater success.
EDIT
@thatguywiththecodes
Yes, THAT Lola was terrible, but as a constructor their history is successful compared to the other cars listed here, with a pole position and 3 podiums to their credit.
@cube
The overall Minardi record may not be spectacular, but they did have a period in the late 1980s/early 1990s when it looked like they had got the hang of F1. In late 1989 Minardi stopped being a joke - I remember when Martini qualified 5th at Estoril and shocked the whole of F1. He then worked further miracles in Spain and Australia, and carried that form into 1990 - second on the grid in Phoenix - staying in the upper midfield for another 4 years. Minardi were crippled by the same thing that did for Pacific and Simtek among others: the huge changes in safety rules after Imola 1994 brought with them a huge financial burden, and for small teams the continual rule changes were simply too much. That Minardi survived at all is worth praising.