tl;dr- No, $77/week isn't a realistic figure for full-time minimum wage workers. There's not really much to debunk here since the figure's source doesn't claim that it's factual or even realistic.
Note that the original website isn't hosted anymore; going to the same URL seems to lead to some sort of blog/advertising site. The below analysis is primarily based on the "challenge toolkit" PDF.
The $77/week figure
As noted in the question, the figure's calculated as the weekly income, less taxes, less living expenses, coming to $78.46/week. As explained by "LIVE THE WAGE CHALLENGE TOOLKIT":
Q: Why $77 a week?
A: The weekly budget for the minimum wage challenge is based on the earnings of a full-time worker earning the federal minimum wage of $7.25 an hour, minus average taxes and housing costs. Tax costs are based on a national average that includes federal and state taxes. Housing costs are based on the national average rent for a 1 bedroom apartment.
So, they're assuming that someone earning minimum wage is renting a 1-bedroom apartment at the average national rate. This isn't at all realistic; most minimum wage workers live with others, and there's simply no reason that someone struggling to buy food wouldn't tend to prefer cheaper apartments.
However, the people issuing the challenge didn't claim that this was realistic in the first place:
Q: Is this a realistic exercise?
A: We have made a number of simplifications to make the Challenge more accessible. For example, we have not factored in geographic variations in taxes and housing costs, and we are not asking participants to consider things like a health care emergency or a broken down car. But the choices you will face in
taking the Live the Wage Challenge are real – these are the sorts of choices that minimum wage workers are confronted with every day.
Further, while the challenge focuses on how hard it is to buy food on that budget, the people pushing the challenge acknowledge that this population's supplemented by food stamps:
Q: Don’t many minimum wage workers receive SNAP benefits to help with their food budget?
A: Yes, minimum wage workers make so little that many receive SNAP (food stamp benefits). In fact, raising the federal minimum wage would lift approximately 3.1 million to 3.6 million people out of SNAP.
This statement itself is misleading, most (76%) people who make minimum wage aren't poor, and most (91%) poor people don't consistency work at all. (reference)
It's a political message, not a factual claim
The document starts with a quick description of the challenge, but most of it's about how to spread the political message. Its table of contents:
- Instructions for the Challenge
- Message Guidance
- Social Media Resources
- Key Facts
- State by State Impact of Raising the Minimum Wage
- Questions and Answers
Ultimately, the $77/week isn't realistic, nor does the source claim that it is.
Its primary problems are:
Most people who earn minimum wage don't live on their income alone. (reference)
Most people who earn minimum wage don't pay for their own single-bedroom apartment (reference), much less at average cost since low-income households spend a fraction of the average (Figure 5).
For the minority of minimum wage earners who are poor, there are food stamps and other assistance programs. (acknowledged by the challenge source)
Poverty's a horrible thing, and this answer isn't meant to suggest otherwise. Merely, the $77/week challenge is political bunk, not factual reality.