English
Is “railroad crossing” meant with 踏切{ふみきり} here?
Yes, but if I say exactly the meaning of "踏切{ふみきり}" in "鳴{な}りっぱなしの踏切{ふみきり}", it is "踏切{ふみきり}の警報音{けいほうおん} sound of the warning signal of the railroad crossing (of Japan)".
マンションのドアを開{あ}けたり閉{し}めたりする音{おと}や鳴{な}りっぱなしの踏切{ふみきり}、しゃべる自動{じどう}販売機{はんばいき}、つけっぱなしのテレビ、駅{えき}のアナウンスなどが、初{はじ}めは本当{ほんとう}にうるさくてたまらなかった。
When I read the original text presented by the questioner (OP), I can say that the above sentence posted by OP was told by a wife from a certain quiet country who had been living in the city of Japan for a somewhat long time.
The sentence presented by OP is an impression of a person who had come to Japan from abroad this time, but this expression is similar to the body of a novel which is expressed as summarized as "都会{とかい}の喧騒{けんそう} the hustle and bustle of a city or the characteristic noises of big cities", where people express the feeling being disgusted to the urban life at the noise.
When interpreting "鳴りっぱなしの踏切" literally, it becomes "a railroad crossing where the alarm does not stop ringing" because the trains go continually just like the crossing called "開かずの踏切 a railroad crossing whose gate never to open" introduced by naruto's comment.
However, judging from the whole sentence presented by OP, it is not literally like "the railroad crossing where the alarm sounds forever", but I think it is "the railroad crossing where the traffic of the trains are so frequent that the alarm sounds constantly or so often but you feel as if the alarm sounded forever."
EDIT
I've never been to japan, do the railroad crossings really constantly ring there? Or do they emit another type of signal, like "blinking"?
Separately from the answer about "っぱなし" in "鳴りっぱなし", I noticed that I did not provide information on railroad crossings in Japan to OP.
So, I will explain about railway crossings in Japan based on the sources written at the end of the explanation and my experience along with the explanation using a picture.
For better understanding of the explanation, I recommend you to look at the picture below at first, then watch the video and read my explanation.
Grade crossing signals are installed at railway crossings also in Japan.
They are the electronic warning devices for road vehicles.
The basic signal consists of flashing red lights, a crossbuck and a bell, attached to a mast. Furthermore, except for crossings with a small traffic volume, striped barriers for the gates are installed. At most crossings, the signals will activate about 40-50 seconds before the train arrives and about 7 seconds before the gates start to be lowered.
The gates will be fully lowered 15 to 20 seconds before the train arrives. The gates will rise or the signals will shut off once the end of the train clears the island circuit.
According to the Ministry of Land, Infrastructure, Transport and Tourism, there are in total about 33,300 level crossings (踏切 fumikiri) in Japan as of 2016. These are easily identifiable with their yellow and black crossbucks mounted adjacent to the crossing, and newer crossings are often paved in green asphalt for easy recognition. Most of these are protected with electronic signals (踏切警報機 fumikiri keihouki) usually equipped with alternating flashing red lights, a bell (note: bell is edited by me) and yellow-and-black-striped barriers. Many signals are also equipped with signs with red LED arrows that indicate the direction of approaching trains.
Similarly to United States school buses, but unlike many other countries, all cars and bicycles must stop before proceeding over any level crossing in Japan, regardless of whether there are electronic signals, as required by the Road Traffic Act. The only exception is if the crossing is additionally controlled by a traffic light, called a fumikiri shingo (踏切信号); in this case, if the light is green, it is not necessary to stop at the level crossing.
On some busy rail lines, especially in urban areas like in Tokyo, Osaka and Nagoya, so many trains pass through some level crossings that they are almost always closed to vehicular traffic. In some cases, such as the Chūō Main Line, more than 50 trains pass in an hour, which equates to only two minutes in which vehicles can cross the tracks during that interval, causing serious traffic congestion and inconvenience. Many such crossings, known in Japanese as akazu no fumikiri (開かずの踏切 a railway crossing whose gates never to open; note: the translation is made by me), have been eliminated by grade separating rail lines, generally by moving them onto viaducts (高架化{こうかか}) or underground tracks (地下化{ちかか}).
Sources:

日本語
Is “railroad crossing” meant with 踏切{ふみきり} here?
はい。しかし、「鳴{な}りっぱなしの踏切{ふみきり}」の「踏切{ふみきり}」の意味{いみ}を正確{せいかく}に言{い}うと、「踏切{ふみきり}の警報音{けいほうおん}です。
マンションのドアを開{あ}けたり閉{し}めたりする音{おと}や鳴{な}りっぱなしの踏切{ふみきり}、しゃべる自動{じどう}販売機{はんばいき}、つけっぱなしのテレビ、駅{えき}のアナウンスなどが、初{はじ}めは本当{ほんとう}にうるさくてたまらなかった。
質問者{しつもんしゃ}が提示{ていじ}した原文{げんぶん}を読{よ}むと、上記{じょうき}の質問{しつもん}の文{ぶん}は、何{なん}らかの理由{りゆう}で静{しず}かな国{くに}から日本{にほん}に移{うつ}り住{す}むようになって相当長{そうとうなが}く、しかも都会{とかい}に住{す}んでいる奥{おく}さんの話{はなし}だと分{わ}かります。
EDIT(文末から移動)
質問者{しつもんしゃ}が提示{ていじ}した冒頭{ぼうとう}の文{ぶん}は、今回{こんかい}は外国{がいこく}から日本{にほん}に来{き}た人{ひと}の感想{かんそう}ですが、「都会{とかい}の喧騒{けんそう}」に嫌気{いやけ}が差{さ}した人{ひと}が使{つか}う感情{かんじょう}の表現{ひょうげん}として小説{しょうせつ}の中{なか}によくありそうな表現{ひょうげん}です。
「鳴{な}りっぱなしの踏切{ふみきり}」をそのまま解釈{かいしゃく}すると、narutoさんが「開{あ}かずの踏切{ふみきり}」として紹介{しょうかい}している踏切{ふみきり}のように、列車{れっしゃ}の往来{おうらい}が絶{た}えないので、「警報音{けいほうおん}が鳴{な}りやまない踏切{ふみきり}」となります。しかし、提示{ていじ}された文全体{ぶんぜんたい}から判断{はんだん}すると、文字通{もじどお}りの「いつまでも警報音{けいほうおん}が鳴{な}りやまない踏切{ふみきり}」ではなく、「絶えず(または「しょっちゅう」)警報音が鳴っているように感じるほど列車の往来が頻繁な踏切」「列車{れっしゃ}の往来{おうらい}が頻繁{ひんぱん}なので警報音{けいほうおん}が鳴{な}りっぱなしと感{かん}じるほど絶えず(または「しょっちゅう」)警報音が鳴っている踏切{ふみきり}」という意味{いみ}だと思{おも}います。
質問者が提示した冒頭の文は、今回は外国から日本に来た人の感想ですが、「都会の喧騒 the hustle and bustle of a city or the characteristic noises of big cities」に嫌気が差した人が使う感情の表現として小説の中によくありそうな表現です。