I workout at home and only have access to dumbells and a barbell. Is there any exercise I can use that hits all the muscles that a pull up does?

Barbell rows, T bar rows, and dumbbell rows are fantastic exercises for back, and you can do them all with just that amount of equipment. The vast majority of people like to use machines for back but it doesnt build the mass nearly as well as free weights. Just make sure to be very picky about your technique and how much weight you use, the biggest mistake people make with free weight back exercises is to rock back and forth too much and use momentum instead of muscular contraction. A test you can use to see if you are using too much weight is in the middle of a set do a rep and hold it at the top. If you cant hold the weight at the stop of the contraction for a second or two you are going too heavy.

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Pain is Temporary, Pride is Forever

you can do every single exercise with a barbell and dumbell. The only equipment I ever touch is a barbell only.

You can't find a place to do a pullup? I bet there are a dozen places around your home you could do them. Pullup bars are overrated.

components of a safe, effective exercise program are Frequency, intensity, and time or duration.
Do not accelerate your workout time too rapidly. If you are overweight and out of fitness.....

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Chicago Fitness Trainer

The purpose of a machine or cable exercise to work a muscle by shifting the way gravity acts on it, a dumbbell and barbell always work downwards since its "free" weight. Cable and machines allow for very specific angles of resistance that are useful for developing back. But if there was a choice to be made, a pure free weight back regimen destroys a machine/cable only regimen any day.

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Pain is Temporary, Pride is Forever

Definitely agree with this. Go find yourself some monkeybars at the park or something! Pull up is by far the best exercise.