What is the best tool for cutting large lumber lengthwise?
I initially totally misread this and was thinking the pieces were 2.4 millimeters thick, not 2.4 meters long. To that end, I've removed comments about hand-sawing - that's... impractical for this type of operation. Power tools are the appropriate answer here.

I'd recommend a band saw for resawing operations like this. Generally, band saw blades take a thinner kerf than a table saw will, so if you need each piece to be as close to the 62.5mm as possible, this would probably be your better bet.  I believe you'll find it easier to find finer teeth on a band saw blade than a table saw blade, and for cutting a piece that's only 2.4mm thick, you're definitely going to want something with very fine teeth if you're looking for a reasonably nice finish - 2.4mm is in the "thick veneer" range.

Upon my reread of the dimensions in question, you will not be too terribly concerned with a fine-tooth, narrow blade for kerf minimization. You'll want a wide blade for straighter tracking, and few, big teeth for fast cutting in tough wood.

Additionally, you want a 250mm (~10") deep cut. On a normal table saw with a 10" blade (fairly standard in the US, not sure about other locales), you won't actually be able to cut all the way through this depth. A band saw will be your only option. You'll need a big band saw that can handle that depth of cut.

The hand-held circular saw mentioned in the comment to this answer: 


Will not have the depth of cut you need. 


The blade is 190mm in diameter, you'll only be able to cut with the 95mm radius, but you don't get full access to that - the depth of cut is only 66mm according to the specs on the page linked.
flipping the sleeper over will only get you a partial cut on the other side.
You'll still have plenty of wood (250mm - 66mm - 66mm = 118mm) left in the middle that you'll have to cut another way. You're only going to cut 1/4 of the way through your sleeper with each pass!

Will not have the control necessary for getting an accurate cut. The sole plate of the circular saw will only have 62 mm (~2.5") to sit on the wood - it will be way too easy to tip the saw, ruining the vertical cut you're trying to make


You don't indicate how many of these pieces you've got to cut, but you might actually want to consider using a hand saw for this - you're only cutting through a 2.4mm depth. A Japanese-style saw can be had with very fine teeth, and for that thickness thinness of material, you'll find that the cut will go pretty quickly by hand.

Also, don't forget that you'll want to measure and cut to take into consideration the kerf if you need both pieces to be the same to the 10th of a mm.

In either powered-saw case (band saw or table saw, as mentioned in another answer), be sure to take great care that you don't lose any fingers (this remains as generally good advice). - you're working with some fairly small pieces...




You will need a band saw to resaw these sleepers. 
You're dealing with pressure treated lumber, so they will be heavier than non-treated lumber
You'll want to make in-feed and out-feed tables to help support the work.


All-in-all, you may want to consider asking the sawmill if they can resaw them for you - they'll have the proper tools and the know-how to do this quickly and safely. Don't just "ask" them to do it, but be prepared to pay for the service - it will be far cheaper than investing in a band saw that will have the capacity you need. Unless, of course, you want a new toy tool. ;)