It is sometimes difficult to distinguish calibration and correction in remote sensing, because we are not in a laboratory with full control on the measurement. Therefore the two are often mixed.
Sensu stricto, radiometric calibration is the conversion from the sensor measurement to a physical quantity. In remote sensing, the sensor is measuring a radiance from the top of the atmosphere. Therefore the image provider also provide calibration coefficients to convert from digit number (DN) to radiance. Because we can trust the amount of light energy that comes from the sun, the radiance is often normalized into a reflectance values (easier to work with because bounded by 0 and one), so this step can also be part of the calibration. So the calibration gives you a reflectance value, but it is the reflectance on top of the atmosphere (TOA).
Indeed, the proportion of the incident light that is really reflected by the observed object is effected by different factors (mainly topography and atmospheric thickness). The reflectances measured TOA therefore need to be corrected if you need absolute values. This does not depend on the sensor itself, so I would not talk about calibration in this case: you need to correct the values measured TOA in order to estimate the values top of canopy.
To answer your question, I would thus say that DOS is a correction method and DN to TOA reflectance is a calibration. DOS require a stable dark object where you can assume that variability is due to atmospheric noise, which is difficult to find.
EDIT: for more info on Landsat atmospheric correction, I recommend LEDAPS (Masek et al, 2013)
For Sentinel-2, different algorithms have been proposed and I cannot give a definitive answer yet. SEN2COR is used a lot, and MAJA is great if you work with time series (also for Landsat, by the way).
It looks like surface reflectance data is what you are looking for. It's been atmospherically corrected so the consistency between different images is much higher. One thing i'd like to remind you is that if you are going to use not only Landsat 8 but also other Landsat surface reflectance data (eg Landsat 4/5/7), the algorithms they used to produce these surface reflectance products are different (LaSRC vs LEDAPS). But if Landsat 8 is all you need then you don't need to worry about it.
Best Answer
You can use the semi-automatic classification plugin to do DOS in QGIS. It is a realy nice (user-friendly) and powerful tool that was improved very intesively since last year. It contains a pre-processing section where you can convert DN values to TOA reflectances and perform a DOS by the way.