How much of "Russia" was actually occupied by the Germans in World War II?

In round figures, the Germans occupied about 700,000 square miles of the Soviet Union in 1941 (at their peak). Of this, roughly 500,000 square miles represented former Soviet Republics: the Baltic states, Belarus, the Ukraine, etc. Another 100,000 represented territory occupied by the Soviets between 1939-1941: eastern Poland, and parts of Romania and Finland. The remainder, about 100,000 square miles, were part of modern day Russia.

This compares with 1.5 million square miles for "European" Russia (in today's terms), so the Germans occupied 6%-7% of European Russia at their 1941 high point. This was somewhat more for the summer/fall of 1942, but the Germans quickly lost most of those gains.

The "buffer states" and the "border regions" did provide a high, though imperfect degree of protection for Russia against German invasion/occupation.

Sources: Various Wikipedia "country" articles on land area.

The map given in Matt's answer shows what the Germans really occupied. The remaining question is what it really means Russia or "Russian territories". I think Tom Au had to specify this exactly when he was asking this question. But I can comment on this.

Double click on the map and you will see the administrative boundaries within Soviet Union. They are shown like this ----- and these are the boundaries between the "Soviet republics" as of 1939.

So if the question is how much of the Russian Federation (in the boundaries of 1939) the Germans occupied, the answer is also clear from the map.

Since then the boundaries of the Russian Federation changed. So modern Russia does not coincide with 1939 Russian Federation. The largest changes are Kaliningrad region (former East Prussia, historically German territory, shown in brown) which was annexed from Germany, and Crimea which was transferred to Ukraine, and then invaded and annexed by Russia in 2014.

But the question is more complicated/ambiguous than that, because Russia itself contains the so-called "autonomous republics" and "autonomous regions" where a great part of population, sometimes majority, is not ethnically Russian. The boundaries of these regions are not shown on this map. This mostly concerns North Caucasus, the place where the Germans penetrated furthest to the East.

The "Russian" cities shown on the map are the following: (NW to SE) Petrozavodsk (Karelia autonomous republic), Pskov, Demiansk, Novgorod (on the boundary of the occupied zone), Rzhev, Vyazma, Smolensk (disputed between Russians and Belorussians, Poles and Lithuanians for centuries) Bryansk, Orel, Kursk, Belgorod (on the very boundary of Ukraine), Voronezh (on the boundary of the occupied zone).

Then go (SE of Ukraine) Rostov, Krasnodar, Kerch (the region where Don Cossaks live, once all this was Ukraine) and Crimea peninsula (populated by Tatars, Ukrainians and Russians, currently occupied and annexed by Russia from Ukraine), Elista (autonomous Republic of Kalmykia, now part of Russia, in 1939 part of Russian republic), Stavropol (North-Caucasus Krai, with very mixed population. The large occupied region around Stavropol consists of 6 autonomous republics: Kalmykia, Dagestan, Chechen Republic, North Ossetia–Alania, Kabardino-Balkar Republic and Karachay–Cherkess Republic).

All other cities in the occupied territory shown in this map are in Baltic republics, Belorussia, Ukraine, and Moldowa.

Visually the pink part which is in the Russian republic of 1939 is about 1/4 to 1/3 of the whole pink part. But the regions occupied by the ethnic Russians is probably 1/5 of the pink area.

maximum extent in December, 1941

It wasn't really the "maximum" extent. The territory, which the USSR fought back in winter 1941 - spring 1942, was much less than the area occupied by the German army while advancing to Stalingrad. The following map clearly demonstrates this.

The "yellow" area is what the USSR gained. Germany's gain is on the south between the two front-lines: Voroshilovgrad (now Lugansk, Ukraine), Rostov-na-Donu, Krasnodar, Stavropol, Elista, etc. - all belong to the RSFSR (and modern Russia).

Also the losses of 1941 included more than just "the road to Moscow". These were large cities (of the RSFSR) such as Smolensk, Belgorod, Bryansk, Kursk, Orel, Pskov etc.

Is it true that occupied Russia was "small" compared to the rest of the occupied Soviet Union?

The administrative borders are seen on the map. These areas are more or less comparable. Though the loss of the RSFSR is mostly "thin" and "stretched" from north to south.

Also it's worth noting that although the total occupied area was only about 7% of USSR, yet the total number of the pre-war population of these territories reached about 40% of the whole USSR (Roughly speaking, the pre-war population of Ukraine+Belorussia+Baltics was about 40-45 mln.; the pre-war population of the occupied parts of the RSFSR 30-35 mln.).

Russia gained troops, civilians, and factories that were withdrawn from the non-Russian part of the Soviet Union in anticipation of German occupation to compensate for lost territory

The Western front's troops were utterly defeated in 1941. Actually many armies were created anew (i.e. regular divisions were disbanded, and newly formed volunteer divisions got their ids).

Factories of Ukraine and Belorussia (mostly) could not be evacuated due to fast enemy advance. What the SU evacuated successfully were factories of the European part of the RSFSR (including the Moscow industrial region). They went as far as Ural, which was a big time loss for the SU. And factories still remaining relatively close to the front were constantly under Luftwaffe's attacks.