I know this is old, but the answer from Summer is dangerously wrong.
Vacuum packaging does not stop bacteria from growing. There are whole groups of bacteria that grow in the absence of oxygen. These are known as anaerobic bacteria, and this category includes some of the more dangerous pathogens; particularly intestinal/gut bacteria, such as the Clostridium species, C. botulinum (causes botulism), C. perfringens (causes vomiting, diarrhoea, enteritis) and C. difficile (causes serious, long-lasting bowel problems known as colitis; very very difficult to treat). Salmonella and Escherichia coli (so called E. coli you may have heard of) can grow in both aerobic and anaerobic situations and also cause serious illness and food poisoning very commonly. There are many others too; the list above is not by any means exhaustive.
Here's what the UK Food Safety Agency has to say:
Vacuum packaging (VP) and modified atmosphere packaging (MAP) can increase the shelf-life of chilled foods by limiting the growth of microorganisms. However, under certain circumstances, a bacterium called non-proteolytic Clostridium botulinum (C. botulinum) may grow in the absence of oxygen. This bacterium is able to grow and produce a harmful toxin at temperatures of 3°C and above.
It is important that VP/MAP chilled foods have appropriate controls in place to minimise the risk of this organism growing and producing harmful levels of toxin.
And the Minnesota Department of Agriculture have to say:
Reduced oxygen packaging (ROP) is intended to extend the shelf life of food products by preventing the growth of most spoilage organisms. To do this, a package reduces oxygen content to a level below what is normally found in the atmosphere. These benefits don’t come without risk, which is why a Hazard Analysis and Critical Control Point (HACCP) plan is required. When oxygen is removed from a package this creates an anaerobic environment which can support the growth of harmful bacteria such as Clostridium bolulinum and Listeria monocytogenes
C. bolulinum toxins can cause severe damage and even lead to death. There is no way to destroy the toxins if they form in food. Listeria monocytogenes may lead to listeriosis, which can be a serious disease for immune compromised individuals and pregnant women, leading to death or still births. Due to the risks associated with ROP, a HACCP plan is required to mitigate harmful bacteria growth or toxin formation.
Examples of reduced oxygen packaging include vacuum packaging, cook-chill, and sous vide.
The good news is, with proper handling by a well trained butcher, the incidence of contaminations of these sorts is low, especially if the gut is not pierced during butchering as this is where a lot of the bacterial species mentioned are commonly found.
The package swelling does NOT necessarily mean that there was oxygen trapped inside. Gas is a product of most mechanisms of respiration and could indicate growth of any species of contaminant. If the package is bubbly or swollen. throw it out! However, absence of swelling is not a reliable indicator that nothing is wrong.
True obligate anaerobes (e.g. Clostridium species) generally create some pretty stinky byproducts, so if your meat does not smell good after opening the packaging, then you should throw it out, as some of the bacterial toxins are heat stable and will survive cooking. Not all bacteria smell to this extent, so this isn't a reliable indicator.
As to how long you can store vacuum packaged meats. The UK Food Standards Agency has a page on vacuum packaging, which if you read the first PDF linked on the page titled The safety and shelf-life of vacuum and modified atmosphere packed chilled foods regarding non-proteolytic Clostridium botulinum indicates a maximum of 13 days. The USA FDA will have similar guidance. However, note that these guides ONLY apply if you store the meat between 3-8 degrees Celsius and this cold chain is maintained. If you have let the packaged meat warm more than 8 C (46.4 F), then it should be frozen and used immediately after thawing or used immediately when not frozen.
Most domestic fridges are more than capable of cooling to less than 8 C and should do so if used properly (i.e. not over-filled, vents not blocked, door not opened a lot).
Your best bet is to ensure that your meat is kept cold as this inhibits bacterial growth more than anything else. Freeze immediately if you can, if not, ensure you keep it unopened and cold and consume within the time indicated on the packaging or immediately after opening.