Usually to extract juice from vegetable, we throw the vegetable into the blender. So, are there other ways that we can extract juice?
Is it that you don't have access to a blender, or you don't like other effects it has in the process of extracting juice? Pretty much any effective method is going to mangle the vegetables and not leave any better or worse an end product in the de-juiced vegetables.
Assuming it's lack of a blender: If they aren't soft and juicy enough to squeeze out juice, pulverizing them with a meat tenderizing hammer while they're in a freezer-strength baggie (strong enough to prevent vegetable shrapnel from flying all over your kitchen) would probably net you at least a few good gulps.
Besides using a juicer, which seems to be the standard answer to other questions in regard to extracting juice (see here and here for examples) there is one method that involves minimal equipment and produces clear juice with a pure, pronounced flavor. It does take a good deal of time, and doesn't have great yield, but it's a shockingly effective method.
I first found this in Kevin Liu's Craft Cocktails at Home, which is sort of a Modernist Cuisine-esque take on cocktails and mixology. He refers to the method as "cryo-juicing", which I like for its very futuristic sound, even though it's dead simple. The basic concept is straightforward: freeze, thaw, and repeat.
Some Introductory Science:
Good old H2O is a very interesting substance. The nature of its hydrogen bonds produces a bond angle of 107.5 degrees and a number of other curious properties (such as polarity, which allows water to dissolve many other substances). Notably, this angle doesn't combine or pack well into regular solids. Where other materials will solidify (i.e. freeze) into regular crystal formations, water's bond angle won't let it pack neatly; as it freezes, the molecules stack into jagged, irregular structures, producing a lot of microscopic spikes and sharp edges. The difference looks a bit like this (even though this shows glass on the right side, the concept is the same):
In food, those irregular edges mean that when water crystals form, they damage and tear your food on a microscopic level from the inside-out. That's why vegetables with a low moisture content tend to do best in the freezer (as stated in this answer) and why meat tends to get mushy if frozen, thawed, and re-frozen (as stated here). It's also why most every food will tend to lose moisture when frozen: water crystals poke tiny holes in cell walls, causing moisture to leak out and evaporate in a dry environment like the inside of a freezer. Interestingly, it's also why ice floats in liquid water.
Cryo-juicing makes this effect work for you, by forming ice crystals to puncture tiny holes in your food and releasing its juice. A juicer or blender uses mechanical action to pulverize your food and destroy cell walls; this is a similar principle, but much subtler. The overall structure of the cell walls is kept intact, but there's just enough damage to extract much of the juice.
Here's what I've found to be the best method:
You can do this with just about any volume you want, but freezing/thawing will take longer with larger amounts. If you want to produce a large quantity of juice, you're best off using multiple smaller bags than a single large one.
Also, you don't have to toss the drained solids; depending on the fruit or vegetable, I'll often steep them with water in a pitcher to extract more flavor. This is particularly nice with cucumbers.
Advantages and Disadvantages:
One of the main things I like about this method is that it produces very clear juice with few suspended particles. This makes for a clean flavor and texture as compared to juice from a blender or juicer. Mechanical action produces tomato juice; cryo-juicing produces something closer to tomato water. And there's no special hardware required, other than inexpensive plastic bags, which unlike a juicer you don't have to clean.
This method does require a lot of time and patience. It can take several hours for each cycle of freezing and thawing, so this isn't something you can prepare quickly. As compared to mechanical juicing, it also has a relatively low yield. At best, you're going to get 50-75% of the volume that you would from mechanical juicing, depending on the food. As an example, a single large cucumber generally produces around 8 oz of cryo-juice, compared to around 12 oz when using a juicer.
Cryo-juicing works best with relatively high-moisture fruits and vegetables with plenty of structural cell walls (i.e. nothing too soft). Moisture content means more water to freeze and puncture, and structure helps with squeezing and extracting juice. Ideal fruits include apples, pears, and melons; good vegetables include cucumber and tomato.
It's possible to cryo-juice softer, fleshy fruits like berries and stone fruits (peaches, plums, etc.) but these tend to break down into mush, and it can be difficult to filter out the juice. Use a fine-mesh strainer if you want to try.
Citrus fruits can also be cryo-juiced, but the results are similar to squeezed juice and squeezing is far more convenient.
Cryo-juicing doesn't work for hard, low-moisture vegetables like carrots, other root vegetables, or hard squash; these really require mechanical action to pulverize their cell walls in order to extract juice. Whether you'd actually want to drink squash juice is another question.
I use this method constantly and with great success. It's low-mess, highly effective, and delicious. I'm an enthusiastic home bartender and find that cryo-juice is an outstanding component in sodas, cocktails, and punches alike, as it brings a very clean flavor. It's also very useful in cooking for applications like poaching or in chilled soups. I won't be throwing out my juicer anytime soon, but I sure don't have to clean it as often.