Breathing would be a very real problem.
According to NASA’s “educational” site on Mars, the Martian atmosphere is 95.32% carbon dioxide, and only .013% Oxygen. Since that site also lists Mars air pressure as 8 millibar (6 mmHg)  compared to Earth’s (1000 millibars = 760 mmHg). This is a tiny fraction of a sparse atmosphere.
On Earth: 21% O2 * 760 mmHg = 160 mmHg of oxygen
On Mars: 0.013% * 6 mmHg = .00078 mmHg
Hemoglobin, the oxygen carrying molecule in blood, is more than just a simple carrier. It’s affinity for oxygen (it has 4 interacting binding sites) is actually higher in low metabolism, high oxygen environments like the lungs, and lower in more acid, CO2 rich, oxygen depleted environments like metabolically active muscle or tissues. In stead of carrying oxygen like a train carrying boxes, it’s like one of those coal cars tha tactually swivels and dumps its coal over the side where needed.
However, .00078 mmHg is far less oxygen than your tissues, and Hemoglobin is still a passive transport (e.g. train cars can’t ‘magnetically’ pull coal up out of the mine, the coal has to flow downhill into them), and its cooperativity curve says that it basically wouldn’t bind any Oxygen at such a low partial pressure.This would only get worse as the lungs themselves started suffocating (becoming acidic and oxygen starved) and therefore decrease hemoglobin’s affinity [trying to force it to dump oxygen it doesn’t have - which would interfere withits ability to pick up oxygen]
There are other serious complications in breathing the Martian atmosphere. For example, Mars has 95.32% carbon dioxide (vs 0.0314% on Earth) This would decrease hemoglobin’s affinity for oxygen (impeding it’s ability to load up on oxygen) and carbon dioxide is our primary repiratory drive (i.e. our brain senses the level of CO[sub]2[/sub], not O[sub]2[/sub], to decide if it needs to breath harder, so you’d gasp like a goldfish) and our primary pH buffer; too much CO[sub]2[/sub] throws off blood pH
Worse, the level of carbon monoxide (.007%) is half the level of oxygen. Carbon monoxide is a potent toxin that binds hemoglobin more readily [and more firmly] than oxygen does. It also has other undesirable physiological effects. This level of carbon monoxide would be steadily toxic in our own oxygen rich atmosphere. On Mars, it’s ruthlessly deadly.
A good science fiction novel may posit a back pack “concentrator” that scavenges the oxygen from the Martian atmosphere to feed an airtight face mask (simple, bulky and inefficent concentrators are already common: people who need supplemental oxygen at home often use them at home instead of tanks). However, less good SF often assumes that “everything can be miniaturized”, but the facemask is necessary to keep out undesired gases,a nd There’s no getting around the fact that you need to suck in over 7500 liters of oxygen to get 1 liter of oxygen. You can’t ‘miniaturize’ that kind of airflow.
Water may not boil in your blood, but it’s still a huge problem for breathing Martian air. Mars water vapor pressure is 0.03% * 6 mmHg = .0018 mmHg. Humans have problems breathing for long periods at a relative humidity of 4% or more, but Mars is 1000x drier! Your lings sinuses and throat would dry out badly, as well as bleeding precious body water into the thirsty Martian atmosphere. To counteract that, you backpack oxygen scavenger would either need a supply of expendable water for humidification, or some sort of water extraction and recycling device [Long duration divers often lose 8-10 lbs in a single work session: compressed air tanks are also very dry air]
1. Erratum: That site’s figure for Martian air pressure (8 millibars = 6 mmHg) is one decimal point lower than the site I used in my last post. If the lower figure is correct, Martian pressure wouldn’t, by itself, be enough to prevent blood boiing and all that, but it still shouldn’t be a problem, between internal pressure sources like blood pressure, and pressure retention by the structural strength of your connective tissue (think of your skin as a tight pair of spandex jeans)