Why should Christians strive to live long lives? At what lengths should they go to do so?

To members of a denomination that believes all a Christian need do is accept Jesus to be saved, live a "good" life, and then go to Heaven, the length of life does seem relatively unimportant.

But other denominations believe that there is a greater purpose to life, and living longer can help with that purpose. (Living longer means looking after one's health; it doesn't mean artificially prolonging the physical life of someone that is terminally ill or mentally unaware.)

The agricultural references throughout both the Old and New Testaments, and the Biblical holy days, symbolically represent God's plan for mankind.

The purpose of converted Christians is to eventually be reborn as literal sons of God, spiritual beings in the same image as Jesus:

And we know that all things work together for good to them that love God, to them who are the called according to his purpose. For whom he did foreknow, he also did predestinate to be conformed to the image of his Son, that he might be the firstborn among many brethren. -- Romans 8:28-29.

Jesus was "the firstborn among many brethren". Creating other beings exactly like him is the goal of Christianity. God sows the seeds, and reaps the harvest. At the last judgement, symbolically the wheat will be separated from the chaff, which is burnt to ashes:

And another angel came out of the temple, crying with a loud voice to him that sat on the cloud, Thrust in thy sickle, and reap: for the time is come for thee to reap; for the harvest of the earth is ripe. -- Revelation 14:15.

People have a human spirit that is more than the spirit of life possessed by animals. At baptism, a new Christian receives part of God's holy spirit, which combines with their own human spirit. Analogous to the union of egg and sperm at physical conception, this combined spirit is the beginning of a new spiritual being.

It is the duty of a Christian's physical body to nurture this new life until it can eventually be born as a spirit being. That means developing character, practising faith, learning to follow God's will, etc.

Some will develop slowly, and others more quickly, but the longer and healthier a life one lives, the more time and opportunity there is to develop this Godly character.

In Matthew 24, it says:

42 “Therefore keep watch, because you do not know on what day your Lord will come. 43 But understand this: If the owner of the house had known at what time of night the thief was coming, he would have kept watch and would not have let his house be broken into. 44 So you also must be ready, because the Son of Man will come at an hour when you do not expect him.

45 “Who then is the faithful and wise servant, whom the master has put in charge of the servants in his household to give them their food at the proper time? 46 It will be good for that servant whose master finds him doing so when he returns. 47 Truly I tell you, he will put him in charge of all his possessions.

The Lord will either return while a person is alive, or upon their death. So long as we are able to work for him, we should do so. However, we should also consider that a person who endures hardship - such as old age - with faith and hope is doing God's work. Even from a hospital bed, a person may still be about the Lord's business.

Many Christians believe that the Lord set a limit of 120 years as the maximum lifespan of a human in Genesis, a decree that began gradually to take effect after the flood. If this is true, then you need not worry about living for centuries before you leave this world.

Genesis 6:

6 When human beings began to increase in number on the earth and daughters were born to them, 2 the sons of God saw that the daughters of humans were beautiful, and they married any of them they chose. 3 Then the Lord said, “My Spirit will not contend with humans forever, for they are mortal; their days will be a hundred and twenty years.”

There are secular accounts of people living beyond 120 years, but all of them also have detractors who claim that the documentation is lacking; e.g. the person's birth certificate was lost in a war or a fire. The Guinness Book of World Records and some other organizations investigate all such claims, and only those with clear documentation are certified. No certified example of a person living over 120 years yet exists. See https://en.wikipedia.org/wiki/List_of_the_verified_oldest_people which has some footnotes about the disputes. This includes Jeanne Calment, the 122 year-old woman, whose daughter disputed her claim.

A commenter asks: "So is simply keeping the Faith a form of fufilling our calling while on earth?"

I do not want to generalize. God requires much of some and little of others. Remember that the wages for all are the same - the one who endured the heat of the day and the worker who joins the crew an hour before the workday ends (See Matthew 20). A personal anecdote shall suffice.

I visited a woman named Janet in her nursing home many years ago. She suffered rheumatoid arthritis, was confined to her bed and endured great pain. She asked our church to send people to pray with her, and when her unbelieving family visited her, she spoke of her faith in God to them. During my visit, I prayed, read scripture and left her a cassette of Christian music.

After I left the facility, I was overcome by tears and wept more bitterly for her than I had ever cried in my life, but a half hour later I was filled with incredible joy, which lasted for a week. Janet died six months later. At the funeral, another of her friends said something with which we all agreed, "When you were with Janet, it was like the Holy Spirit was pouring out of her." That is what keeping the faith meant for Janet, and what it did for me and others. Though she was dying, she did for me - as her work for God - what I had never been able to do for myself. She set me free from over a decade of depression.