I have a Wikipedia-level knowledge of Russian history in the 18th and 19th centuries. My sense is the Peter the Great and his successors tried to modernize Russia economically and militarily and met with mixed success at best.
Nevertheless, Russia seemed quite successful in imposing its will on the Ottoman Empire, which was also trying to modernize with mixed results. In addition, Russia was able to serve as "Europe's policeman" and go toe-to-toe with modern European military powers in a way that other less-modernized entities around the world (the Turks, Persians, Indians, Chinese, etc.) could not. Yet, by World War I Russia was in bad enough shape that it couldn't arm all of its own troops.
This is admittedly a broad question, but to summarize: Russia seemed to be in the same boat as a lot of other non-Western powers in the 18th and 19th centuries, in terms of modernity (and the power that comes with it). However, Russia seems to have been a lot more powerful militarily than the other non-Western powers.
Why is this? Are one of my premises wrong?
Russia wasn't feudal in its organization at all. In fact Russia's aristocracy was deliberately designed by Peter to be dependent on state service for social status (although that gets more complicated later). What Russia had was an infinite supply of serf or state peasant recruits--in effect, mirroring some of the advantages that universal conscription would bring to Western and Central Europe after the French Revolution--and an industrial base that was entirely built around the needs of the army. In other words, relative to the size of the economy, Russia wasn't very industrialized, but in the industries where it counted (low-grade fabric for uniforms, weapons, munitions) the state spent an enormous amount of money, time, and attention making sure they were world-class. It wasn't always enough, but a lot of the time it was.
(And, yeah, World War I didn't quite go how you describe it--there were plenty of weapons, enough to fight a civil war with afterwards, but the leadership was poor.)
In very short: Paul Kennedy writes about this in his book The Rise and Fall of the Great Powers. He explains this by stating that although Russia wasn't as developed as the Western states, its army with its modern weapons were still far superior to the nomadic armies of the Asians plains, so its expansions wasn't as difficult as it would seem.
"Yet, by World War I Russia was in bad enough shape that it couldn't arm all of its own troops"
I wouldn't say it is completely true. Russia mobilized far faster than it had been expected by the Central Powers before the war, although badly lead, the Russian Army was a formidable foe.
Follow-up Question:
Why was Russia able to keep up in military technology, whereas other large/rich/powerful countries that had decent contacts with Europe for a long time (Ottomans, Safavids, Indians) fell behind in military technology?