I think you might be a little unclear on what money actually is or why we use it. Physically, money may be just a few scraps of paper, metal discs, or numbers on a spreadsheet. When you say it shouldn't matter what something "costs" you seem to invoke the idea that all we need to do is change some numbers in a spreadsheet to solve world poverty. In reality, money is a concept and bill, coins, spreadsheets are just tokens for keeping track of it and what it represents, the scarcity of resources.
There are all kinds of things that it is physically possible to do but it isn't physically possible to do all of them once. The cost of something represents the available supply of that something as well as demand for it. It determines what things are done and who gets what in a fair manner. Or at least it would in perfect world. I could go on about why markets fail in certain circumstances and how they should be regulated to nudge them closer to the ideal models but that's not terribly relevant. The important thing is that most of the time most of the cost of something is determined by the resources spent in making it, what else those resources could have produced instead, how many people want that something, and how much they want that something.
As long as there is scarcity I can't imagine any alternative to actual money that isn't essentially a new name for money (like "spending" karma points). Barter is completely impractical in the modern world of specialized jobs (How many malware infections do I have to remover for a new pair of headphones and where can I find someone who needs those services and has the headphones I want?) and our standard of living isn't possible without those specializations. In a world with limited resources the only other way to eliminate money would be for the government to plan every last detail of every person's life down what groceries the get each week and the toys their kids get for Christmas. No government can even properly run a command economy on macro scale so I think it's safe to say that working out such tiny details would be a failure.
Combining money with a market economy farms out all those decisions to individuals and makes a large economy practical. Money is just any arbitrary object that we've decided as a society we will all accept as barter. It allows for specialization. I can barter my skill in fixing computers to my employer for money and then barter that money for things I need or want but which my employer doesn't have. With some people cheating the system and others falling through the cracks it's hardly perfect but like democracy and the scientific method it's the best we've got so far. The laws of supply and demand are just as much a part of reality as the law of gravity and we have to take them into account when planning an economic system just like we take the force of gravity into account when designing a building.
Even without money, something that is scarce and desirable will be worth something even if that worth is not expressed in dollars and cents. People will barter for it, take risks to get it, or make whatever sacrifice is necessary in order to attain it and those trades, risks, will be in proportion to their assessment of its worth.
I'm all for reforming our economic system so as many people as possible can lead fulfilling lives and have as few economic worries as possible but barring the invention of a completely concept to replace it, the idea of eliminating all money is literally the very last step on the list of thing to do in order to achieve that goal.