Yes.
Yes, you can two Immovable Rods as an infinite ladder or monkey bars, and it's really cool.
Immovable Rod
This flat iron rod has a button on one end. You can use an action to press the button, which causes the rod to become magically fixed in place. Until you or another creature uses an action to push the button again, the rod doesn’t move, even if it is defying gravity. The rod can hold up to 8,000 pounds of weight. More weight causes the rod to deactivate and fall. A creature can use an action to make a DC 30 Strength check, moving the fixed rod up to 10 feet on a success.
It's certainly reasonable to try holding each rod in one hand in such a way that your thumbs are on the buttons, so that you can alternately release and engage the rods. As a rod is released, you could swing forward holding onto the fixed rod, and engage the free rod, thereby propelling yourself across the moat. You could do the same thing to climb vertically, although you'd be doing a one-handed pull-up as you moved each rod. The GM might impose strength, dexterity, or even constitution checks along the way. Depending on the distance and other factors, fatigue might become a serious issue. Particularly annoying is the description of it being a flat iron rod. Wrapping it with leather to make handgrips might be more comfortable.
There might be easier ways.
If you don't want to cling to the darned things while you're trying to push the buttons on the ends, you can tie a rope and a stirrup to each. Then stand in the stirrups and alternately activate the rods.
No matter what, it's going to be slow, because it'll most likely take a turn per activation. It could take a long time to go very far.
A little invisibility might help with getting spotted as you dangle by your arms over the shark-infested moat picked out by spotlights.
Up here in the castle we've got boxes of those things. We resell 'em down in the thieves' quarter. You wouldn't believe the number of would-be anti-heroes who try to get over the walls that way. Sometimes we get several a night. We generally roll dice as to whether to use fireball, heat metal, or grease. Sometimes we just lasso them. If it's slow we use a mage hand to mess with the buttons.
Then there was the guy who was so strong he didn't bother with pressing the buttons. He'd just keep pulling on the things til he'd get them to move. That dude was no fun at all once we got him on deck. Serious attitude problem.
Then there was the assassin who got past us somehow and nailed a visiting duke. We think she was using those rods, but we're not sure. She must have had some other trick up her sleeve.
Yeah, between the yahoos with the immovable rods, the flyers, the spider climbers, the monks, the thieves, the polymorphed lizards, the familiars, the invisible flying carpets and everything else, it gets pretty busy up here. One night we had so many we almost ran out of charges on the gems of seeing.
Telekinesis, a level 5 spell, says you can manipulate an object. I would think that the example of opening a door is at least as complex as pushing a button on a rod.
You can exert fine control on objects with your telekinetic grip, such as manipulating a simple tool, opening a door or a container, stowing or retrieving an item from an open container, or pouring the contents from a vial.
Also it does not say that you cannot manipulate a magical object. So I think this one is valid.
Unseen Servant, a level 1 spell, should also be able to do it.
The servant can perform simple tasks that a human servant could do, such as fetching things, cleaning, mending, folding clothes, lighting fires, serving food, and pouring wine.
Again, there is nothing saying that it cannot activate a magical item. (However, it would not be able to read a scroll or drink a potion)
However, you say at a distance and the Unseen Servant needs to be able to reach that location. Some people say the Unseen Servant can hover, others say it cannot.
Ten Foot Poles could also be used to push on a button. You do not specify the distance, but there are no reasons why that would not work. Only it may take a few tries (i.e. Dexterity check on a fairly high DC such as DC 16.)
Missiles could also be thrown at it (rocks, arrows, bolts, etc.) However, that may be quite a feat to hit that tiny little button (although a giant can throw big rocks, he! he!)
Best Answer
I believe that while the bag and rod would be locked in place, an Immovable Rod halfway into a Bag of Holding will still deactivate or be moved with enough weight or force. That includes any weight in excess of 8,000 lbs. (4 tons) or a DC 30 Strength check. Note that the effects of extreme weight and extreme strength on an Immovable Rod are different.
The text of Immovable Rod is:
If more than 8,000 lbs is placed on the Immovable Rod, even if it's placed halfway in a Bag of Holding, it deactivates and the rod and bag fall. A strength check could have multiple results. A creature can pull the rod from the bag (causing the bag to fall off), push it further into the bag (leaving it in the bag's pocket dimension, meaning the bag will fall), or push the rod and the bag 10 feet (the rod stays halfway in the bag). Also consider that weight put on the button could cause it to shut off immediately, even if it's less than 8,000 lbs.
Up to that point, however, I would rule that any force placed on the rod is no different with or without the Bag of Holding. Even placing a ton of weight (2,000 lbs) on the rod shouldn't puncture or break the bag, as the rod is fixed in space and supports the weight entirely. However, I wouldn't say there's any benefit in terms of stability gained by attaching a Bag of Holding to one end, and neither does the linked answer. It simply says that attempting to move an active Immovable Rod into a dimensional space in a Bag of Holding would require a strength check to do, as it's considered to be moving the rod.
Keep in mind that the linked answer is not written rule, merely an interpretation of the effects of an Immovable Rod, the pocket dimensional space of a Bag of Holding, and the nature of physics in the D&D system. It does not override the rules of an Immovable Rod, unless the DM decrees it so.