Then what was Russia's aim in the Russo-Japanese War? Aside from Manchuria.
Correct me if I'm wrong, but didn't the Russians send their fleet halfway around the world in 1904, only to have it destroyed the next year?
You keep mentioning 1904. That's obviously a very different question since while by then the Russians could clearly maintain considerable force on the Pacific, there were no more easy pickings for a really profitable base there, except what they already had overland in Siberia. If they'd have hung on to Alaska, well that's about equivalent to their options in Siberia (Alaska has oil? Gold? Guess what, so does Siberia!) and hanging on there is again a POD long before 1904. It seems to me what you'd want would be either a much earlier POD as your choice of forum indicates, where the capabilities of the steam dreadnoughts of 1904 and what logistic support they could provide overland in their eastern ports by then are irrelevant, or you are asking the much tougher question of how well could they muscle in on the existing hegemonies of the Pacific in the early 20th century.
Or possibly a mid-to-late 19th century question, WI instead of selling Alaska (which they did OTL because they wanted money fast because they didn't have a lot of it) they invested a lot more (that they didn't think would be cost-effective OTL) in reinforcing their bases in the Maritime province and Alaska and went a-Viking for warmer acquisitions to the south, before it was all pre-empted by strong European powers? Say they decided to beat Japan to the punch in Taiwan or capitalize on Spanish troubles in the Philippines?
Again, it would be very costly to develop their North Pacific holdings to sustain such efforts. I'm not sure if Juneau or Sitka can in any sense qualify as winter open-water ports, I rather doubt it. So again their problem is either to secure something more suitable along the north Asian coast (and somehow supply it overland, or even develop it into a working shipyard) thus running straight into trouble with not only Japan but the British, neither of whom would want them pre-empting Korea or Chinese ports, and all that as a clear preliminary to challenging these same powers and yet others in the more tropical zones of the Pacific, not to mention the USA that would be feeling its oats around this post-Civil War era. Or send an expedition clear around the world from the Baltic or Black Sea, through the gauntlet of all their enemies and allies of potential enemies, right past Spain either way (a likely target of their ambitions). Well before 1900 at least the prevailing standard would not be steamships and so they wouldn't face the problems their Black Fleet did in getting coal on the way, but they'd be that much slower and sailing past all sorts of choke points. Such as around the Cape of Good Hope past British squadrons, or risking their life and limb across the straits of South America where the British also were swarming, or past Singapore not to mention the entire Indian Ocean just waiting for the British to intercept them there...
Basically if you can see a way the British might favor such an enterprise instead of opposing it, maybe then they'd only be facing the sheer daunting cost of sending enough of an expedition to win something worth winning and holding by such roundabout ways, so far out of communication with their capital. But I don't see it.
If they can beef up their holdings on the Northwest Pacific enough to launch such expeditions, they'd probably prefer to concentrate on using such good logistics to take more of North China and/or Korea overland instead. And while I can think of some warm Pacific targets they might then want to go after--Hawaii, Taiwan, the Philippines--all would be hotly contested by other powers with more established power in the region--Britain foremost, but also Japan and the USA, and the longer they delay the more solid their claims get.
Trying to go for it earlier, you get the objections others have raised. 1904 just doesn't seem relevant, except as a demonstration that even then Russian power on the high seas
in the Pacific was marginal and second-rate, however strong they might have been in European waters. It's just so difficult to project sufficient logistics overland to support such adventures from a Pacific base or to send it the long way round without having first acquired strong bases farther west.
Hmn, maybe if the Russians early on took and held the South American straits, being better able to endure the climate there than anyone else but say Norwegians? But the British were keen on keeping that passage open for themselves from quite early on; again we'd want an unlikely and long-lasting Anglo-Russian alliance.
Or if you like, totally ignore the question of the advantage their holdings on the Siberian coast give them initially, and ask whether the Russians could have been players in the Atlantic-based age of sail conquests on the high seas, getting bases like the South American passage or South Africa--the latter would then require yet more acquisitions in the Indian Ocean to secure a route to the Pacific, whereas straight out from Tierra Del Fuego to Pacific plums is a long open ocean passage. But maybe the Russians get these the same way England, France, the Dutch got theirs? They are still likely to lose out to Britain in the long run, but maybe they can first acquire something worth hanging on to in the southern Pacific and then switch over to keeping contact via the Siberian ports?
I guess then they would hang on to Alaska even if it were unprofitable in itself, to shield their Siberian bases and access to their valuable Pacific colony.
So, focus on the 16th and early 17th century and ask how plausible was a strong Russian Atlantic navy and how much Russia had in the way of colonizing potential, and how likely they were to get the alliance of one of the major maritime powers of that period. Say, they help the Dutch sustain themselves (but I think OTL they tended to ally with France in this time frame) or team up with the French to collectively hold the balance against Britain, and knowing they are reaching the North Pacific overland let themselves be steered by the French into taking their chances on finding or grabbing something in the Pacific via the South American/Antarctic straits. Or somehow throw their lot in with the British and are awarded a shot at these prizes?