The Championship side were due to face Hull in a sixth round tie on the weekend of August 22/23.
Eagles, who won the 1895 Cup at Wembley last August, have also announced they will not be competing in this autumn’s one-off competition for Championship and League One clubs.
Their announcement came a week after the Rugby Football League confirmed the lower division seasons have been abandoned.
Sheffield are among five non-Super League teams through to the sixth round of the Cup, which could now be redrawn.
A club statement said: “The club shares the bitter disappointment that will no doubt be felt by our supporters, partners, and sponsors in that we will not see the Eagles in action again in 2020.”
But it stressed: “These steps have been taken to help safeguard our long-term future.
“The RFL’s decision to declare the 2020 Betfred Championship null and void we believe was the right one.
“Our governing body consulted with clubs rigorously over the past few months in what has been an ever-changing landscape.
“There was no decision that could have been made that would have satisfied all clubs and [we] understand the frustration expressed by some, but the work the RFL carried out to try and return was thorough with the decision reflecting feedback from clubs across both divisions.”
Eagles say they withdraw from the Challenge Cup on “player welfare” grounds.
The statement added: “Despite our thanks going to the RFL for their continued hard work we are disappointed in their decision not to further support those non-Super League clubs who remain in the Coral Challenge Cup.
“Player welfare was the overriding factor in exiting this competition.
“Super League sides will have played three rounds before the next stage of the Cup was due to take place compared to over five months since our last competitive fixture.
“It was felt it wasn't safe to compete and to do so would risk serious injury to our players.
“As a club we proposed that the ‘prize money’ for the losing sides exiting the competition in the next round, the sixth round (£7,500) and the quarter-fiinals (£15,000), should have been split equally.
“This would have resulted in the losing clubs in both the sixth and the quarter-finals receiving £11,250 each, a significant increase for our club.
“The tie would have been the club’s first home match against a Super League side since 2015 and would have generated revenue significantly larger than the above figure, but our proposal was declined.”
The RFL plan to stage a competition for lower division clubs in October and November, but entry is not mandatory.
Eagles’ statement insisted: “Whilst the proposed autumn competition would allow spectators to return to matches, the financial burden placed on the club to do so would have been severe, including but not limited to Covid-19 testing, additional training and match day costs and removing our 22 players and two staff from the government’s furlough scheme.
“Costs would have far outweighed any income generated and our utmost importance is the welfare of our players which we felt could have been compromised should we have competed in either competition.”
Eagles say pre-season training, for their 2021 Championship campaign, will begin in November.
The club have urged fans who can afford it not to request a refund on 2020 season tickets.