So the thing about the Eastern Han (which directly turned into the Three Kingdoms) was that it wasn't all that militarized in terms of having massive professional standing armies. The Eastern Han Dynasty specifically did not have large standing armies as it didn't want to recreate the chaos of the earlier Western Han Dynasty where powerful regional rulers (princes in those cases) had huge standing armies with which to threaten the centre.

The Eastern Han's actual state military was comprised of two parts: the standing "Northern Army" of 4,000 professional soldiers split into five regiments (at an early point in his career the warlord and later chancellor Cao Cao was colonel of one of the five northern army regiments, and IIRC the later warlords Yuan Shao and Yuan Shu all held posts in the Northern Army at some point) which were in charge of guarding the capital (the province of Sili, which began in the east at around Capital Luoyang and stretched to the west to around Chang An) and the vast "Southern Army" which were disparate volunteer armies raised by regional governors, inspectors, protectors and such who could be raised in times of need (such as during the time of the Yellow Turban rebellion). But they weren't standing armies as, being that their loyalty was to the local gentry and governors, their loyalty was not directly to the court. This system worked as it was supposed to, even the Yellow Turban rebellion was put down with huge success in a short amount of time, and did not lead to any real problems for the stability of the Emperor.

The real problem came when Dong Zhuo's army (which was a small, professional standing army that guarded the north-western borders) was invited into the capital by Yuan Shao (btw, if we had to pinpoint a single person for the fall of the entire Eastern Han Dynasty I'm pretty sure it'd be Yuan Shao, he's not a stupid man in general but in this case he made a monumentally stupid mistake) to deal with the eunuch faction. The rest as you might know is history: the eunuch faction killed He Jin, head of the Empress' household in retaliation, and in response He Jin's followers such as Yuan Shao and Cao Cao purged the Eunuch faction from the capital. What you had in this case was mutual destruction of the two power structures (the Empress' clan and the Eunuchs) who was essentially propping the Emperor up (as their authority derived from the Emperor, all other authority can derive from large gentry families, local governors, border generals etc.). Without any real powerful institutions remaining loyal to the Emperor, Dong Zhuo was very easily able to sieze power with quite a small army of just initially 3,000 (he did this by apparently marching his men into the city, secretly taking them out at night and marching them back in, giving the Capital the impression of a huge army bearing down which the Northern Army could not face - not that they wanted to, they essentially had no command structure by this point due to the aforementioned power struggle between the Empress/Gentry faction and the Eunuch faction).

The rest is of course history: the governors and large families from the Eastern provinces militarized and rose up to oppose Dong Zhuo in a coalition, they fought to a bloody draw, but without any semblance of Imperial authority and with no institutions left in real support of the (child) Emperor, nobody could safely disband their armies, so the regional governors and such stayed militarized. Power devolved to the local governors and gentry etc. until we just had a fractured "Empire" of warlords, and thus began the Three Kingdoms.

Source:

"Pin Sanguo" Yi Zhongtian
"Chronicles of the Three Kingdoms" Chen Shou, annotations by Pei Songzhi
"Later Han Military Organisation" Rafe de Crespigny